tensor products of cuntz semigroups associated with

Tensor products of Cuntz semigroupsassociated with Choquet simplices

Masterarbeitzur Erlangung des akademischen Grades

Master of Science

Westfälische Wilhelms-Universität MünsterFachbereich Mathematik und Informatik

Maximilian Stoffel

Münster, September 2018

Erster Gutachter: Dr. Hannes Thiel

Zweiter Gutachter: Prof. Dr. Wilhelm Winter

Contents

Introduction 1

1 Cu-semigroups 41.1 The category Cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The Cuntz semigroup of a C∗-algebra . . . . . . . . . . . . . . . . . . . . . . . 61.3 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Inductive limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Compact convex sets 172.1 Partially ordered abelian groups with order unit . . . . . . . . . . . . . . . . . . 192.2 Approximations of (semi-)continuous functions . . . . . . . . . . . . . . . . . . 232.3 LAff(K)0

++ is a Cu-semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Choquet simplices 323.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Continuous and affine functions on Choquet simplices . . . . . . . . . . . . . . 353.3 Bauer simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 A tensor product for compact convex sets and the main theorem 454.1 The tensor product of groups with order unit . . . . . . . . . . . . . . . . . . . 454.2 Dimension groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 A tensor product for compact convex sets . . . . . . . . . . . . . . . . . . . . . 484.4 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 The Cu-semigroup L(F (S)) 605.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 The realification of a Cu-semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 The range of the natural map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4 The image of the functor LAff(_)0

++ . . . . . . . . . . . . . . . . . . . . . . . . 66

6 An application of the main theorem 72

Bibliography 75

Introduction

The Cuntz semigroup is an invariant for C∗-algebras. It was introduced by Cuntz in [Cun78] asa means of proving the existence of so called dimension functions on unital, simple and stablyfinite C∗-algebras. He showed that a unital, simple C∗-algebra is stably finite if and only ifit admits a 2-quasitracial state (which in the presence of exactness is the same as a tracialstate). The Cuntz semigroup proved to be a fine invariant when Toms (in [Tom08]) gave acounterexample to a version of the Elliott conjecture which asserted that the class of all unital,separable, simple, nonelementary, nuclear C∗-algebras is classified by the Elliott invariant,an invariant for C∗-algebras of K-theoretic nature. He gave an example of two C∗-algebraswith the aforementioned properties that cannot be distinguished by a number of invariants,including the Elliott invariant, but which are nonisomorphic because their Cuntz semigroupsdiffer. Today, thanks to the work of numerous mathematicians, it is known that smaller classesof C∗-algebras are classified by the Elliott invariant. For example, when substituting ‘nuclear’for the stronger condition ‘finite nuclear dimension’, the above version of the Elliott conjectureholds true.Subsequently, in [CEI08], Coward, Elliott and Ivanescu introduced the category Cu of abstractCuntz semigroups. Objects in Cu are ordered monoids that satisfy a number of additionalaxioms. They showed that a slightly modified version of the Cuntz semigroup of a C∗-algebraA, denoted by Cu(A), always belongs to Cu. The category Cu is not only a useful frameworkfor the study of Cuntz semigroups of C∗-algebras, but it is also interesting in its own right. Forinstance in [APT18], Antoine, Perera and Thiel established the existence of tensor productsin Cu (relative to a certain notion of Cu-bimorphism). It is then only natural to ask howCu(A⊗ B) and Cu(A)⊗ Cu(B) are related, for C∗-algebras A and B (say for simplicity thatA is nuclear). It turns out that there is a natural morphism Cu(A) ⊗ Cu(B) → Cu(A ⊗ B),which under certain circumstances is known to be an isomorphism (for example, if A or B isan AF-algebra), but which in general is not an isomorphism.The Cuntz semigroup of a sufficiently regular C∗-algebra A can be expressed as the disjointunion of two semigroups: the Murray-von Neumann semigroup V (A) and the lower semicon-tinuous affine functions on the compact convex set QT1(A) of all normalized quasitraces onA. In general, for every metrizable compact convex set K, the lower semicontinuous affinefunctions on K that are either strictly positive or zero, LAff(K)0

++, form a semigroup inthe category Cu. One would expect LAff(QT1(A))0

++ ⊗ LAff(QT1(B))0++ to be a component

1

Introduction

of Cu(A) ⊗ Cu(B), which leads us to the following question: what is the tensor product ofLAff(K1)0

++ and LAff(K2)0++, for two metrizable compact convex sets K1 and K2? While the

answer to this question remains unknown, we provide an answer in the special case that eitherK1 or K2 is a Choquet simplex. The main result of this thesis states that there exists anisomorphism

LAff(K1)0++ ⊗ LAff(K2)0

++∼= LAff(K1 ⊗K2)0

++

whenever K1 and K2 are two metrizable compact convex sets, one of which is a Choquetsimplex. Here K1 ⊗ K2 denotes the so called biprojective tensor product of compact convexsets, a concept that is known since the late 1960s.This thesis is organized as follows: in the first chapter, we recapitulate a few general aspectsof the theory of Cuntz semigroups. Beginning with the definition of the category Cu and theconstruction of the Cuntz semigroup of a C∗-algebra, we discuss how certain properties ofa C∗-algebra are encoded in its Cuntz semigroup. We give an overview of the constructionof tensor products and inductive limits in the category Cu. At the end of the chapter, wesee that a large class of simple Cu-semigroups can be expressed as the disjoint union of twosubsemigroups: the so called compact part and the so called soft part.In the second chapter, we are concerned with compact convex sets. Two important classesof examples of compact convex sets are examined, both of which will show up constantly:probability measures on compact Hausdorff spaces and state spaces of partially ordered abeliangroups with order unit. A rough duality between the category of compact convex sets and thecategory of partially ordered abelian groups with order unit is developed. This duality allowsus to define a tensor product for compact convex sets later on. We prove a few approximationresults - most notably that any lower semicontinuous affine function on a compact convexset can be approximated by continuous affine functions. This allows us to demonstrate thatLAff(K)0

++ is a Cu-semigroup whenever K is a metrizable compact convex set.Then, in the third chapter, we concentrate on Choquet simplices, i.e. compact convex sets witha certain property that ensures a large supply of continuous affine functions. The subclass ofBauer simplices is discussed. We are particularly interested in how the property of being aChoquet simplex or a Bauer simplex is encoded in the (lower semi-)continuous affine functions.Before we arrive at the main result in the fourth chapter, we take a look at a certain notionof tensor products for compact convex sets. We verify that the tensor product of two Choquetsimplices (Bauer simplices, metrizable compact convex sets) is a Choquet simplex (a Bauersimplex, metrizable). One difficulty in the proof of the main theorem is overcome by applyinga structural theorem of Effros, Handelman and Shen concerning inductive limits of finite powersof Z in the category of partially ordered abelian groups.In the fifth chapter, we determine which Cu-semigroups are of the form LAff(K)0

++, for somemetrizable compact convex set K. Our argumentation relies on a theorem of Robert that gives

2

Introduction

a concrete description of the so called realification of a Cu-semigroup.In the final chapter, we briefly discuss how the main theorem could be applied to obtain a newcriterion for which the natural map Cu(A)⊗ Cu(B)→ Cu(A⊗B) is an isomorphism.

As for the notation, we use N to denote the set of all natural numbers, including zero. If wewant the zero to be excluded, we will write N∗. If x1, . . . , xn, y1, . . . , ym is a finite collectionof elements in some partially ordered set (X,≤), the expression x1, . . . , xn ≤ y1, . . . , ym shallmean that xi ≤ yj holds for all i ∈ {1, . . . , n} and j ∈ {1, . . . ,m}.

I would like to thank my supervisor Dr. Hannes Thiel for introducing me to Cuntz semigroups,for his support, the frequent discussions on the development of this thesis with its occurringobstacles, and overall for successfully guiding me through this project.

3

1 Cu-semigroups

In this chapter, we recall a few results concerning Cu-semigroups. Our general reference forthis chapter, especially for the first two sections, is [Thi16].

1.1 The category Cu

Let x and y be elements in some partially ordered set (X,≤) (from now on denoted by X). Wesay that x is way-below y, x� y, if for every increasing net (zi)i∈I in X that has a supremumsuch that y ≤ supi zi, there exists some index i0 such that x ≤ zi0 . Similarly, we say thatx is sequentially way-below y, x �ω y, if for every increasing sequence (zn)n∈N in X thathas a supremum, we have that y ≤ supn zn implies that there exists some index n0 such thatx ≤ zn0 . The expression ‘increasing net in X’ may be substituted for ‘upward directed subsetof X’. Similarly, ‘increasing sequence in X’ may be changed to ‘upward directed subset ofX admitting a countable cofinal subset’. In general, x � y implies that x �ω y which itselfimplies that x ≤ y. Also, the inequality x ≤ x′ � y′ ≤ y implies that x � y. An analogousstatement holds for �ω. If X has a smallest element 0, then 0 � x holds for all x ∈ X. Therelation � (�ω) is certainly most interesting if every increasing net (increasing sequence) inX has a supremum, which leads us to the following definition.

1.1 Definition A partially ordered set X is called directed complete (sequentially complete)if every increasing net (increasing sequence) in X has a supremum. We abbreviate the termdirected complete partially ordered set to dcpo. Similarly, we refer to a sequentially completepartially ordered set as ω-dcpo.We say that X is continuous (ω-continuous) if for every x ∈ X, there exists an increasing net(xi)i∈I (increasing sequence (xn)n∈N) in X such that xi � x for each i ∈ I (xn �ω x for eachn ∈ N) and such that x = supi xi (x = supn xn). In the sequential case, this is equivalent tosaying that there exists a �ω-increasing sequence (xn)n in X with supremum x.Finally, a domain (ω-domain) is a continuous dcpo (ω-continuous ω-dcpo).

There is a suitable separability condition for which � and �ω agree:

1.2 Definition A subset B of some ω-dcpo X is called basis if for all x, y ∈ X satisfying

4

1 Cu-semigroups

x �ω y, we can find an element b ∈ B such that x ≤ b �ω y. In this case, every elementx ∈ X can be written as the supremum of an increasing (or even �ω-increasing) sequence inB. We say that X is countably based if it has a countable basis.

1.3 Proposition A countably based ω-dcpo is a domain if and only if it is a ω-domain. Inthis case, the relations � and �ω agree.

1.4 Definition A partially ordered set X is called an inf-semilattice if every nonempty finitesubset of X has an infimum. The term sup-semilattice is defined analogously. If X is bothan inf-semilattice and a sup-semilattice, we say that X is a lattice. If any subset of X has asupremum and an infimum, we say that X is a complete lattice.

1.5 Example Let X be a locally compact Hausdorff space, and denote the topology on X

by O(X). When equipped with the inclusion relation, O(X) becomes a partially ordered set.Any subset A ⊆ O(X) has a supremum, which is given by

⋃A, and an infimum, which is

given by the interior of⋂A, so O(X) is a complete lattice. In fact, O(X) is a domain. The

way-below relation is given by U � V if and only if there exists a compact subset K of X suchthat U ⊆ K ⊆ V (for U, V ∈ O(X)). For this reason, the way-below relation is also known asthe compact containment relation. If X is second countable, then O(X) is countably based.

Let M and N be abelian monoids. Recall that a monoid homomorphism from M to N is amap M → N that preserves the zero element and addition. We let AM denote the category ofall abelian monoids with monoid homomorphisms.A positively ordered monoid is an abelian monoidM , equipped with a compatible partial ordersuch that every element in M is positive (i.e. 0 ≤ x for all x ∈M). A PoM-morphism betweenpositively ordered monoids M and N is an order-preserving monoid homomorphism M → N .Let PoM be the category of all positively ordered monoids and PoM-morphisms.

1.6 Definition ([CEI08]) A Cu-semigroup is a positively ordered monoid S that satisfies thefollowing axioms:(O1) Every increasing sequence in S has a supremum.(O2) For every s ∈ S, there exists a �ω-increasing sequence in S with supremum s.(O3) If s, s′, t, t′ ∈ S satisfy s�ω s

′ and t�ω t′, then s+ t�ω s

′ + t′.(O4) If (sn)n∈N and (tn)n∈N are increasing sequences in S, then the equation supn(sn + tn) =

(supn sn) + (supn tn) holds.A generalized Cu-morphism from a Cu-semigroup S to a Cu-semigroup T is a map S → T thatpreserves the zero element, addition, order and suprema of increasing sequences. If this mapalso preserves the sequential way-below relation, then we refer to it as a Cu-morphism. Let Cu

5

1 Cu-semigroups

be the category whose objects are Cu-semigroups and whose morphisms are Cu-morphisms.

Remark Axiom (O1) states that S is a ω-dcpo, while axiom (O2) just means that S is ω-continuous. Thus, axioms (O1) and (O2) say that S is a ω-domain.

1.7 Definition An ideal in some Cu-semigroup S is a submonoid J that is downward hered-itary, i.e. if x ∈ S and y ∈ J satisfy x ≤ y, then x lies in J , and that is closed under supremaof increasing sequences. Let us denote the set of all ideals in S by Lat(S). We say that S issimple if the only ideals in S are the trivial ideals {0} and S.

The set Lat(S) is naturally ordered by inclusion. Since S itself is an ideal and since arbitraryintersections of ideals are again ideals, it follows that for any subset D ⊆ S, there exists asmallest ideal J in S containing D. In this case, J equals the intersection of all ideals inS containing D. It follows that Lat(S) is a complete lattice. If a is an element in S, set∞ · a := supn n · a. It is straightforward to show that the ideal generated by a is given by{s ∈ S | s ≤ ∞· a}. If S is simple and a is nonzero, this ideal must be equal to S. In this case,∞ · a is the unique largest element of S, and we denote it by ∞.

1.8 Definition A functional on a Cu-semigroup S is a generalized Cu-morphism S → [0,∞].We denote the set of all functionals on S by F (S).

1.2 The Cuntz semigroup of a C∗-algebra

Probably the most important reason for the study of Cu-semigroups is that one can assign toeach C∗-algebra A an object Cu(A) in Cu. This Cu-semigroup is known as the Cuntz semigroupof the C∗-algebra A. This assignment defines a functor from the category of C∗-algebras to thecategory Cu, so the Cuntz semigroup can be regarded as an invariant for C∗-algebras. In thissection, we will recall some details of the construction of Cu(A) and how certain properties ofA are encoded in its Cuntz semigroup.

1.9 Definition Let a and b be positive elements in some C∗-algebra A. We write a - b ifthere exists a sequence (zn)n in A such that a = limn znbz

∗n. In this case we say that a is Cuntz

subequivalent to b. If both a - b and b - a hold, we say that a is Cuntz equivalent to b, anddenote it by a ∼ b.

It is easy to see that the relation - is reflexive (use an approximate identity for A) andtransitive. Thus, the relation ∼ defines an equivalence relation on A+.

1.10 Definition The Cuntz semigroup of a C∗-algebra A is defined as follows: As a set, it

6

1 Cu-semigroups

is given by Cu(A) := (A ⊗ K)+/ ∼, where K denotes the compact operators on a separable,infinite-dimensional Hilbert space H. We define a partial order on Cu(A) by setting [a] ≤ [b] ifand only if a - b. Finally, we can define an addition on Cu(A) as follows: Let ψ : K⊗M2 → Kbe any ∗-isomorphism. Then idA ⊗ ψ : A⊗K ⊗M2 → A⊗K is also a ∗-isomorphism. We set

[a] + [b] := [(idA ⊗ ψ)(a⊕ b)],

where a⊕ b :=(a 00 b

)∈ A⊗K ⊗M2.

Remark That the binary relation in Definition 1.10 is a well defined partial order follows fromthe fact that Cuntz subequivalence is transitive and reflexive. However, it is more difficult toshow that the addition is well defined. It is well known that any ∗-isomorphism ϕ : K → K hasthe form ϕ(_) = u · id(_) · u∗ for some unitary u ∈ B(H). If ψ1, ψ2 : K ⊗M2 → K are two∗-isomorphisms, it follows that there exists a unitary u ∈ B(H) such that ψ1(_) = u ·ψ2(_) ·u∗.Then the element u′ := 1A ⊗ u (where A denotes the minimal unitalization of A) is a unitaryin the multiplier algebra of A⊗K and satisfies

(idA ⊗ ψ1)(a⊕ b) = u′ · (idA ⊗ ψ2)(a⊕ b) · (u′)∗,

for all a, b ∈ (A ⊗ K)+. In particular, (idA ⊗ ψ1)(a ⊕ b) and (idA ⊗ ψ2)(a ⊕ b) are unitarilyequivalent with unitaries from the multiplier algebra of A ⊗ K. It follows from [Thi16, 2.20]that [(idA ⊗ ψ1)(a⊕ b)] = [(idA ⊗ ψ1)(a⊕ b)], so this value is independent of the choice of the∗-isomorphism K⊗M2 → K. It is not very hard to show that the value [(idA⊗ψ)(a⊕ b)] doesnot depend on the choice of the representatives a and b either and consequently, the additionon Cu(A) is well defined. Again, one easily sees that the order and addition as defined in 1.10give Cu(A) the structure of a positively ordered monoid.

1.11 Theorem ([CEI08, Theorem 1]) If A is a C∗-algebra, then Cu(A) is a Cu-semigroup.For [a], [b] ∈ Cu(A), we have [a]�ω [b] if and only if there exists ε > 0 such that a - (b− ε)+

(the element (b− ε)+ is defined as f(b), where f : R→ R is given by f(x) := max{x− ε, 0}).Let ϕ : A→ B be a ∗-homomorphism into another C∗-algebraB. The induced ∗-homomorphismϕ ⊗ id : A ⊗ K → B ⊗ K preserves Cuntz subequivalence, and Cu(ϕ) : Cu(A) → Cu(B) is aCu-morphism. The assignment above defines a functor Cu : C∗ → Cu, where C∗ denotes thecategory of all C∗-algebras with ∗-homomorphisms.If A is separable, then Cu(A) is countably based.

1.12 Example The Cuntz semigroup of K can be computed as Cu(K) ∼= N, where N :=

N ∪ {∞} is equipped with the obvious order and addition. The isomorphism is given by[a] 7→ rank(a), for a ∈ K⊗K ∼= K. In general, any C∗-algebra A satisfies Cu(A) ∼= Cu(A⊗K).In particular, we obtain Cu(C) ∼= Cu(K) ∼= N.

7

1 Cu-semigroups

1.13 Definition For a positively ordered monoid S, consider the following axioms:(O5′) For all x, x′, y ∈ S satisfying x′ �ω x ≤ y, there exists s ∈ S such that x′+s ≤ y ≤ x+s.(O5) For all x, x′, y, y′, z ∈ S satisfying x′ �ω x, y′ �ω y and x + y ≤ z, there exists s ∈ S

such that x′ + s ≤ z ≤ x+ s while also y′ ≤ s.(O6) For all x, x′, y, z ∈ S satisfying x′ �ω x ≤ y + z, there exist s, t ∈ S such that s ≤ x, y,

and t ≤ x, z, and x′ ≤ s+ t.We say that S has almost algebraic order if S satisfies (O5). Also, S is said to have almostRiesz decomposition if it satisfies (O6). It is clear that (O5) is stronger than (O5′).

The axiom (O5) has first been considered in [APT18], where it was also shown that the Cuntzsemigroup of every C∗-algebra satisfies (O5). The weaker version (O5′) was introduced byRørdam and Winter in [RW10]. Axiom (O6) is due to Robert in [Rob13]. In the same paper,it was shown that the Cuntz semigroup of any C∗-algebra satisfies (O6).

1.14 Theorem Let A be a C∗-algebra. If I C A is a (closed, two-sided) ideal, then the set

JI := {[a] ∈ Cu(A) | a ∈ (I ⊗K)+}

is an ideal in Cu(A). The map Cu(I) → Cu(A), induced by the inclusion I → A, is anembedding of Cu-semigroups, and the image of the former map is given by JI . Thus, we mayidentify Cu(I) with JI . Conversely, if J is an ideal in Cu(A), then the set

IJ := {a ∈ A | [aa∗] ∈ J}

is an ideal in A.The assignments above establish an order isomorphism Lat(A) ∼= Lat(Cu(A)), I 7→ Cu(I). Inparticular, a C∗-algebra A is simple if and only if its Cuntz semigroup is simple.

Since we can recover the spectrum of a C∗-algebra from the lattice of all ideals, the Cuntzsemigroup is a complete invariant for abelian C∗-algebras.

1.15 Corollary Two abelian C∗-algebras A and B are isomorphic if and only if their Cuntzsemigroups are isomorphic.

1.16 Definition A 1-quasitrace on a C∗-algebra A is a function τ : A+ → [0,∞] with thefollowing properties:• τ(0) = 0 and τ(t · a) = t · τ(a) for all a ∈ A+ and t ∈ (0,∞),• τ(a+ b) = τ(a) + τ(b) for all a, b ∈ A+ satisfying ab = ba,• τ(a∗a) = τ(aa∗) for all a ∈ A.

Let n ∈ N∗. A n-quasitrace on A is a 1-quasitrace on A that extends to a 1-quasitrace onA⊗Mn. Every 2-quasitrace is a n-quasitrace, for all n ∈ N∗ (see [BH82, II.4.1]). Additionally,

8

1 Cu-semigroups

the extension to A ⊗Mn is unique. It follows that every 2-quasitrace extends uniquely to a1-quasitrace on A⊗K. We refer to a 2-quasitrace just as a quasitrace. Let us denote the set ofall lower semicontinuous quasitraces on A by QT(A). A trace on A is a 1-quasitrace on A thatis additive on all elements (not just commutating ones). We use T(A) to denote the set of alllower semicontinuous traces on A. Finally, if A is unital, we use QT1(A) and T1(A) to denotethe sets of all normalized (quasi-)traces on A. We remark without proof that every normalized(quasi-)trace on A is automatically real-valued, order-preserving and continuous. Thus, T1(A)

may be identified with the set of all tracial states on A.

For the upcoming theorem, we use the following notation: for n ∈ N∗, let fn : R → R be thefunction defined by

fn(x) :=

0 if x ≤ 1

n

nx− 1 if 1n ≤ x ≤

2n

1 if 2n ≤ x

.

1.17 Theorem Let A be a C∗-algebra. If τ ∈ QT(A) is a lower semicontinuous quasitraceon A, then the map dτ : Cu(A)→ [0,∞], defined by dτ ([a]) := limn τ(fn(a)), is a well definedfunctional on Cu(A). Conversely, if λ ∈ F (Cu(A)) is a functional on Cu(A), then the functionτλ : (A⊗K)+ → [0,∞], which is defined by the formula

τλ(a) :=

∫ ∞0

λ([(a− t)+])dt,

is a lower semicontinuous quasitrace on A.These assignments define a natural bijection QT(A) ∼= F (Cu(A)).

1.3 Tensor products

Throughout this thesis, we will discuss various different tensor products. All of them, exceptthe tensor product of compact convex sets, are defined relative to a notion of bimorphism. Thefollowing definition is informal, but it should make clear what we mean by a tensor productassociated with a notion of bimorphism, and should therefore avoid repetition.

1.18 Definition Let C be a category whose objects are sets endowed with additional structures(such as addition, order, etc.) and whose morphisms are structure-preserving maps. For objectsA, B, C in C, a C-bimorphism from A×B to C is a map A×B → C (where A×B denotes thecartesian product of the underlying sets) that preserves the structures of A and B in a certainsense that must be specified beforehand. Usually, this map should preserve some structureof A and B in each variable while preserving other structure of A and B jointly. We denotethe set of all C-bimorphisms A × B → C by BiC(A × B,C). If D is another object in C, the

9

1 Cu-semigroups

composition of any C-bimorphism A × B → C with any C-morphism C → D should alwaysresult in a C-bimorphism A×B → D.Let A and B be objects in C. A tensor product of A and B is a pair (C,ω), consisting of someobject C in C and a C-bimorphism ω : A×B → C, such that the following universal propertyis satisfied: For any object D in C and for every C-bimorphism ϕ : A× B → D, there exists aunique C-morphism ϕ : C → D such that ϕ = ϕ ◦ ω.If the tensor product of A and B exists, it is unique up to isomorphism. We write A ⊗C Binstead of C. The universal property may be restated by saying that there is a natural bijection

HomC(A⊗C B,_) ∼= BiC(A×B,_).

For a ∈ A and b ∈ B, we write a�b (for ‘algebraic tensor products’, namely the tensor productsin AM, PoM, PreW, OAG and GOU) or a⊗ b (for the tensor product of compact convex setsand the Cu-tensor product) instead of ω(a, b). An element of this form is called elementarytensor. If the tensor product exists for all objects A and B in C, we say that the category C hastensor products, or that the tensor product of C-objects exists. Note that although the notionof the tensor product heavily depends on the definition of C-bimorphisms, this dependence isneither represented in the terminology nor in the notation of the tensor product. This shouldnot lead to confusion though, since every category in this thesis will have at most one notionof bimorphisms associated with it.

It was shown in [APT18] that the category Cu has tensor products (with respect to a certainnotion of Cu-bimorphism). The general strategy involves regarding Cu as a full, reflectivesubcategory of a certain category PreW. It is easier to construct a tensor product in thecategory PreW since the objects in that category are algebraic in nature. The tensor productin Cu is then obtained by taking a certain completion of the tensor product in PreW.

1.19 Definition An auxiliary relation on a positively ordered monoid S is a relation ≺ thatsatisfies the following conditions:• For all s ∈ S, we have that 0 ≺ s.• For all s1, s2 ∈ S, the inequality s1 ≺ s2 implies that s1 ≤ s2.• If s1, s2, s3, s4 ∈ S satisfy s1 ≤ s2 ≺ s3 ≤ s4, then s1 ≺ s4.

1.20 Definition Suppose that S is a positively ordered monoid, equipped with a certainauxiliary relation ≺. For any element s in S, we define the set s≺ := {t ∈ S | t ≺ s}. Considerthe following axioms:(W1) For every s ∈ S, there exists a ≺-increasing sequence (sn)n ⊆ s≺ that is cofinal in s≺.(W2) Each s ∈ S is the supremum of s≺.(W3) If s, s′, t, t′ ∈ S satisfy s ≺ s′ and t ≺ t′, then s+ s′ ≺ t+ t′.

10

1 Cu-semigroups

(W4) If s, t, u ∈ S satisfy s ≺ t + u, then there exist t′, u′ ∈ S such that t′ ≺ t and u′ ≺ u

while also s ≺ t′ + u′.A pair (S,≺), consisting of a positively ordered monoid S and a fixed auxiliary relation ≺ onS, is called PreW-semigroup if the axioms (W1), (W3) and (W4) are satisfied. A W-semigroupis a PreW-semigroup (S,≺) that also satisfies axiom (W2). If (S,≺) and (T,≺) are two PreW-semigroups, a generalized W-morphism (S,≺) → (T,≺) is a PoM-morphism ϕ : S → T thatsatisfies the following continuity condition: If s ∈ S and t ∈ T satisfy t ≺ ϕ(s), then thereexists an element s′ ∈ S with s′ ≺ s and t ≤ ϕ(s′). If this map also preserves the auxiliaryrelations, then we refer to it as a W-morphism. Let PreW be the category whose objects arePreW-semigroups and whose morphisms are W-morphisms. Then let W be the full subcategoryof PreW whose objects are W-semigroups.

1.21 Example If S is a Cu-semigroup, then (S,�ω) is a W-semigroup. For any other Cu-semigroup T , the Cu-morphisms S → T are exactly the W-morphisms (S,�ω) → (T,�ω).Thus, we may regard Cu as a full subcategory of W, which itself is a full subcategory of PreW.We will write S instead of (S,�ω).

1.22 Theorem ([APT18, 3.1.6]) Let (S,≺) be a PreW-semigroup. There exists a Cu-semigroupγ(S,≺) and a W-morphism α : (S,≺) → γ(S,≺) with the following universal property: Forany Cu-semigroup T and any W-morphism ϕ : (S,≺)→ T , there exists a unique Cu-morphismϕ : γ(S,≺)→ T such that ϕ = ϕ ◦ α.As usual, γ(S,≺) is unique up to Cu-isomorphism. We refer to γ(S,≺) as the Cu-completionof (S,≺). The completion functor γ : PreW → Cu is left adjoint to the inclusion functorCu → PreW (this is just a restatement of the universal property). In particular, Cu is a full,reflective subcategory of PreW.

1.23 Definition Let S, T and U be sets, and let ϕ : S × T → U be a map.1) Suppose that S, T and U are abelian monoids. We say that ϕ is a AM-bimorphism if it is

a AM-morphism in each variable.2) Suppose that S, T and U are positively ordered monoids. We say that ϕ is a PoM-

bimorphism if it is a PoM-morphism in each variable.3) Suppose that S, T and U are PreW-semigroups. We say that ϕ is a W-bimorphism if it is

a generalized W-morphism in each variable and if for all s, s′ ∈ S and t, t′ ∈ T satisfyings ≺ s′ and t ≺ t′, we have that ϕ(s, t) ≺ ϕ(s′, t′).

4) Suppose that S, T and U are Cu-semigroups. We say that ϕ is a Cu-bimorphism if it isa generalized Cu-morphism in each variable and if for all s, s′ ∈ S and t, t′ ∈ T satisfyings�ω s

′ and t�ω t′, we have that ϕ(s, t)� ϕ(s′, t′).

11

1 Cu-semigroups

1.24 Theorem ([APT18]) 1) The tensor product of abelian monoids exists.2) The tensor product of positively ordered monoids exists, more precisely: Let S and T be

positively ordered monoids, and let (S⊗AMT, ω) be their tensor product as abelian monoids.Then there exists a partial order ≤ on S ⊗AM T such that S ⊗PoM T := (S ⊗AM T, ≤) is apositively ordered monoid, such that ω is a PoM-bimorphism and such that (S ⊗PoM T, ω)

is the tensor product of S and T as positively ordered monoids.3) The tensor product of PreW-semigroups exists, more precisely: Let (S,≺) and (T,≺) be

PreW-semigroups, and let (S ⊗PoM T, ω) be the PoM-tensor product. There exists anauxiliary relation ≺ on S ⊗PoM T such that (S,≺) ⊗PreW (T,≺) := (S ⊗PoM T,≺) is aPreW-semigroup, such that ω is a W-bimorphism and such that (S,≺)⊗PreW (T,≺) is thetensor product of (S,≺) and (T,≺) as PreW-semigroups.

4) The tensor product of Cu-semigroups exists, more precisely: If S and T are Cu-semigroups,let ((S,�) ⊗PreW (T,�), ω) be the tensor product of S and T , considered as PreW-semigroups. Set S⊗T := γ((S,�)⊗PreW (T,�)), and let α : (S,�)⊗PreW (T,�)→ S⊗Tbe the PreW-morphism belonging to the completion. Then (S ⊗ T, α ◦ ω) is the tensorproduct of S and T as Cu-semigroups.

Remark Suppose that S, T and U are Cu-semigroups. A map ϕ : S × T → U is called ageneralized Cu-bimorphism if it is a generalized Cu-morphism in each variable. The Cu-tensorproduct of S and T also has the universal property that generalized Cu-bimorphisms S×T → U

correspond to generalized Cu-morphisms S ⊗ T → U .

1.25 Theorem ([APT18, 6.3.5]) There is a Cu-isomorphism

γ((S,≺)⊗PreW (T,≺)) ∼= γ(S,≺)⊗ γ(T,≺)

that is natural in PreW-semigroups (S,≺) and (T,≺).

1.4 Inductive limits

The category Cu has inductive limits. Similarly to the case of the tensor product, this followsfrom the fact that the category PreW has inductive limits. The inductive limit in PreW isbased on the construction of the inductive limit in the category Set of all sets. Let us recallsome details.Suppose that ((Mi)i∈I , (ϕi,j)i,j∈I,i≤j) is an inductive system in Set. For a ∈ Mi and b ∈ Mj ,we write a ∼ b if there exists an index i, j ≤ k ∈ I such that ϕi,k(a) = ϕj,k(b). This definesan equivalence relation on the disjoint union

⊔i∈IMi. Set M := (

⊔i∈IMi)/∼, and define

ϕi,∞ : Mi → M by ϕi,∞(x) := [x]. It is straightforward to check that (M, (ϕi,∞)i∈I) is the

12

1 Cu-semigroups

inductive limit of the inductive system ((Mi)i∈I , (ϕi,j)i,j∈I,i≤j). We refer to this inductive limitas the algebraic inductive limit.There are many examples of categories that have inductive limits which are based on thealgebraic inductive limit. For instance, the category PoM has inductive limits. The inductivelimit is just the algebraic inductive limit, equipped with the induced order and addition. Inthe same vein, the category PreW has inductive limits which is again given by the algebraicinductive limit, equipped with the induced order, addition and auxiliary relation. Since Cu isa full, reflective subcategory of PreW, it follows that Cu has inductive limits. More precisely,if ((Si)i∈I , (ϕi,j)i,j∈I,i≤j) is an inductive system in Cu, the inductive limit is the completion ofthe inductive limit in PreW:

Cu- lim−→Si ∼= γ(PreW- lim−→Si).

1.26 Theorem ([APT18, 3.2.9]) The functor Cu : C∗ → Cu preserves inductive limits.

The above theorem is a useful tool for computing Cuntz semigroups of C∗-algebras arising asinductive limits of simpler C∗-algebras. For example, using Theorem 1.26, one can computethe Cuntz semigroup of the CAR-algebra M2∞ as

Cu(M2∞) ∼= N[12 ] t (0,∞],

where N[12 ] denotes the set of all positive dyadic numbers, and where addition and order are

defined as follows:

a+ b :=

{a+ b ∈ N[1

2 ] if a, b ∈ N[12 ]

a+ b ∈ (0,∞] otherwiseand a ≤ b :⇔

{a < b if a ∈ N[1

2 ], b ∈ (0,∞]

a ≤ b otherwise,

for a, b ∈ N[12 ] t (0,∞]. A detailed computation can be found in [Thi16, 4.33].

1.5 Composition and decomposition

In many cases, a Cu-semigroup consists of two components: a ‘compact’ part and a ‘soft’part. For example, the Cuntz semigroup of M2∞ is the disjoint union of the compact part N[1

2 ]

and the soft part (0,∞]. More general, a large class of simple, stably finite Cu-semigroupsdecomposes into a compact part and a soft part. On the other hand, given a triple (C,D, γ),consisting of certain ordered semigroups C and D and a so called composition map γ : C → D,one can construct a Cu-semigroup C tγ D in which the compact and soft parts are given by Cand D respectively.The composition of semigroups and decomposition of Cu-semigroups was studied in [Eng14]. Inthis section, we will recall some of Engbers’ results. All Cu-semigroups in [Eng14] are assumedto satisfy (O5′) and (O6), so we will add these axioms to our assumptions.

13

1 Cu-semigroups

Let S be a Cu-semigroup. An element s ∈ S is called compact if s �ω s. We denote the setof all compact elements in S by C(S). Clearly, C(S) is a submonoid of S. Let us use D(S) todenote the set of all elements that are either zero or noncompact. For any monoid M , we setM× := M \ {0}.

1.27 Definition ([Eng14, 6.1.1]) We say that a Cu-semigroup S is decomposable if there existsa monoid homomorphism γS : C(S)→ S such that

γS(x) = max{y ∈ S | y ≤ x and y 6= x} 6= 0

holds for each x ∈ C(S)×. If such a map exists, it is necessarily unique. The element γS(x) iscalled the predecessor of x, and the map γS is called the predecessor map of S.

Let C be the full subcategory of PoM consisting of all algebraically ordered PoMs (a PoM M

is algebraically ordered if for all x, z ∈M , we have x ≤ z if and only if there exists y ∈M suchthat x+ y = z). Let D denote the full subcategory of Cu which consists of all Cu-semigroupssatisfying (O5′) and (O6) and that have no compact elements beside the zero element.Suppose that C is an object in C. An ideal in C is a downward hereditary submonoid. If theonly ideals in C are {0} and C, we say that C is simple.

1.28 Definition ([Eng14, 8.1.2]) Suppose that C and D are simple semigroups in C and D.We say that a map γ : C → D is a composition map if the following conditions are satisfied:• For all x ∈ C, we have γ(x) = 0 if and only if x = 0.• γ is additive.• For all x, y ∈ C×, the inequality γ(x)�ω γ(y) implies that x ≤ y and x 6= y.• For all x ∈ C and y ∈ D×, we have γ(x) �ω y if and only if there exists z ∈ D× such

that γ(x) + z = y.• For all x ∈ C× and y, z ∈ D× satisfying γ(x)�ω x+ y, there exist y′, z′ ∈ D× such thaty′ �ω y, γ(x), and z′ �ω z, γ(x), and γ(x) ≤ y′ + z′.

In [Eng14, 8.1.3 and 8.2.1], Engbers demonstrates that there is a one-to-one correspondencebetween the following classes:1) Simple, decomposable Cu-semigroups satisfying (O5′) and (O6).2) Triples (C,D, γ), consisting of a simple semigroup C in C, a simple semigroup D in D and

a composition map γ : C → D.To each Cu-semigroup S as in 1), one assigns the triple (C(S), D(S), γS). Conversely, a triple(C,D, γ) as in 2) gives rise to a Cu-semigroup S := C tγ D, which is defined as follows: as a

14

1 Cu-semigroups

set, S is defined as S := C× tD× t {0S}. Addition and order are defined by

x+S y :=

y if x = 0S

x if y = 0S

x+C y if x, y ∈ C×

γ(x) +D y if x ∈ C×, y ∈ D×

x+D γ(y) if x ∈ D×, y ∈ C×

x+D y if x, y ∈ D×

, and

x ≤S y :⇔

y ∈ S if x = 0S

x = 0S if y = 0S

x ≤C y if x, y ∈ C×

γ(x)�ω,D y if x ∈ C×, y ∈ D×

x ≤D γ(y) if x ∈ D×, y ∈ C×

x ≤D y if x, y ∈ D×

,

for x, y ∈ S. Engbers then shows that this one-to-one correspondence can, when suitablyinterpreted, be understood as an equivalence of categories. As shown in [Eng14, 6.2.4 and 6.3.2],the class of all simple and decomposable Cu-semigroups satisfying (O5′) and (O6) includes thefollowing subclasses:• Cuntz semigroups of simple, separable, nonelementary, stably finite C∗-algebras, and• simple, countably based, nonelementary, weakly cancellative Cu-semigroups satisfying

(O5′) and (O6).A Cu-semigroup S is called weakly cancellative if for all x, y, z ∈ S, the condition x+z �ω y+z

implies that x �ω y. The definition of an elementary Cu-semigroup can be found in [Eng14,3.1.6].

An element s in some Cu-semigroup S is called soft if for all s′ �ω s, there exists some k ∈ Nsuch that (k + 1)s′ ≤ ks. A nice feature of soft elements is that their order is determined byfunctionals, assuming that S almost unperforated (meaning that for all x, y ∈ S, the condition(k + 1)x ≤ ky for some k ∈ N implies that x ≤ y).

1.29 Theorem ([APT18, 5.3.12]) Let S be an almost unperforated Cu-semigroup. Supposethat x and y lie in S, that x is soft and that λ(x) ≤ λ(y) holds for every functional λ ∈ F (S).Then x ≤ y.

1.30 Definition Let S be a Cu-semigroup. An element s ∈ S is called infinite if there existsa nonzero element x ∈ S such that s = s + x. We say that s is finite if it is not infinite, andwe call S stably finite if for all s ∈ S, the condition s �ω s

′ for some s′ ∈ S implies that s isfinite.

15

1 Cu-semigroups

1.31 Proposition ([APT18, 5.3.16]) If S is a simple and stably finite Cu-semigroup thatsatisfies (O5′), then every element in S is either compact or soft. The zero element is the onlyelement that is both compact and soft.

Remark Let S be a simple Cu-semigroup satisfying (O5′). Under the mild assumption that Sis stably finite, it follows from Proposition 1.31 that D(S) equals the set of all soft elements.It is shown in [APT18, 5.2.10] that a nonzero, simple Cu-semigroup S is stably finite if andonly if ∞ is not compact. It follows that S is stably finite if and only if ∞ lies in D(S).A unital, simple C∗-algebra A is stably finite if and only if Cu(A) is stably finite (see [Thi16,6.22]).

The upcoming proposition is taken from [Thi16, 6.40]. Let me explain some notation. We useZ to denote the Jiang-Su algebra. If A is a (unital) C∗-algebra, we use V (A) to denote theMurray-von Neumann semigroup of A. Recall that QT1(A) denotes the set of all normalizedquasitraces on A. It turns out that QT1(A) is a so called Choquet simplex, so we can considerthe set LAff(QT1(A))0

++ of all lower semicontinuous affine functions on QT1(A) that are eitherstrictly positive or zero. Then LAff(QT1(A))0

++ is a particularly nice Cu-semigroup, and thefollowing holds:

1.32 Proposition For a unital, simple, separable, stably finite, Z-stable C∗-algebra A, wehave that

Cu(A) ∼= V (A) tγ LAff(QT1(A))0++,

where the composition map is given by γ : V (A)→ LAff(QT1(A))0++, [a] 7→ (τ 7→ τ(a)).

As you can see, the Cu-semigroup LAff(K)0++ for some metrizable compact convex set K

appears naturally in the study of Cuntz semigroups of sufficiently regular C∗-algebras.

With the help of Proposition 1.32, one can compute the Cuntz semigroup of the Jiang-Sualgebra as Cu(Z) ∼= N tγ [0,∞], with the obvious composition map γ : N → [0,∞], x 7→ x.Other authors prefer to write Cu(Z) ∼= N t (0,∞] for the same Cu-semigroup. On a similarnote, we have seen that Cu(M2∞) ∼= N[1

2 ]t(0,∞]. In Engbers’ notation, this should be writtenas N[1

2 ] tγ [0,∞], with the obvious composition map γ.

From now on, we will use � instead of �ω to denote the way-below relation in a generalCu-semigroup S.

16

2 Compact convex sets

Compact convex sets play a central role in this thesis; so in this chapter, we cover their basictheory. By a (compact) convex set, we always mean a (compact) convex subset of a locallyconvex, Hausdorff, real topological vector space. Up until section 2.3, we closely follow theapproach taken in [Goo86].

Let X be a set, and let f, g : X → [−∞,∞] be functions. We write f ≤ g if f(x) ≤ g(x) holdsfor all x ∈ X. Similarly, the inequality f < g shall mean that f(x) < g(x) holds for everyx ∈ X. For any c ∈ [−∞,∞], we also use c to denote the constant function that takes thevalue c.Now let X be a topological space. Recall that a function f : X → [−∞,∞] is called lowersemicontinuous if f−1((t,∞]) is open for every t ∈ R. Equivalently, for all x ∈ X and for everynet (xi)i in X that converges to x, the inequality f(x) ≤ lim infi f(xi) holds. We say that f isupper semicontinuous if −f is lower semicontinuous.Finally, let K be a convex set, and suppose that f : K → [−∞,∞] is a function such that theimage of f is contained in [−∞,∞) or in (−∞,∞]. We say that f is convex if the inequalityf(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) holds for all x, y ∈ K and t ∈ [0, 1] (where 0 · ∞ and0 · (−∞) are defined to be 0). Also, f is called concave if −f is convex. We say that f is affineif it is both convex and concave, i.e. the equality

f(tx+ (1− t)y) = tf(x) + (1− t)f(y)

holds for all x, y ∈ K and t ∈ [0, 1]. Throughout this thesis, we will consider the following sets:

Aff(K) := {f : K → R continuous and affine},Aff(K)+ := {f ∈ Aff(K) | f ≥ 0},Aff(K)0

++ := {f ∈ Aff(K) | f > 0 or f = 0},LAff(K) := {f : K → (−∞,∞] lower semicontinuous and affine},LAff(K)+ := {f ∈ LAff(K) | f ≥ 0},LAff(K)0

++ := {f ∈ LAff(K) | f > 0 or f = 0}.

We equip LAff(K) with pointwise order and addition. Observe that LAff(K) is a dcpo, wherethe supremum of an increasing net is just given by the pointwise supremum. Taking supremaof increasing nets is compatible with addition.

17


In the first half of this chapter, we will build a sufficiently large supply of intersesting examplesof compact convex sets.Let X be a compact Hausdorff space. A probability measure on X is a Radon measure µ onX satisfying µ(X) = 1 (we refer to [Arv96] for details). We denote the set of all probabilitymeasures on X by M+

1 (X). According to the Riesz-Markov theorem, the assignment

µ 7→ (f 7→∫Xfdµ)

defines a bijection

M+1 (X) ∼= {ϕ : C(X;R)→ R positive, linear, ϕ(1) = 1} ⊆ C(X;R)∗,

where C(X;R)∗ denotes the topological dual space of the continuous, real valued functionson X. From now on, we will identify M+

1 (X) with the set of all normalized positive, linearfunctionals on C(X;R). For x ∈ X, let δx denote the dirac measure at x. It is defined byδx(B) = 1 if x ∈ B, and δx(B) = 0 if x /∈ B, for any Borel set B. Note that the dirac measureat x is the unique probability measure on X satisfying δx({x}) = 1.

2.1 Proposition Let X be a compact Hausdorff space. When equipped with the weak-∗

topology, C(X;R)∗ is a locally convex, Hausdorff, real topological vector space and containsM+

1 (X) as a compact convex subset.

Proof. (cf. [Goo86, 5.22]) It is clear that M+1 (X) is convex and closed in the weak-∗ topology.

Let ϕ ∈M+1 (X). Any f in the unit ball of C(X;R) satisfies −1 ≤ f ≤ 1, which implies that

−1 = ϕ(−1) ≤ ϕ(f) ≤ ϕ(1) = 1,

hence the norm of ϕ is bounded by 1. It follows that M+1 (X) is contained in the unit ball of

C(X;R)∗ which is weak-∗-compact by the Banach-Alaoglu theorem. It follows that M+1 (X) is

in fact a compact convex set.

2.2 Proposition For any compact Hausdorff space X, the assignment x 7→ δx defines ahomeomorphism X ∼= ∂eM

+1 (X).

Proof. (cf. [Goo86, 5.24]) Given x ∈ X, let us show that δx is extreme. Consider a convexcombination δx = tµ1 + (1− t)µ2, where µ1, µ2 ∈M+

1 (X) and t ∈ (0, 1). We have that

1 = δx({x}) = tµ1({x}) + (1− t)µ2({x}),

which is only possible if µ1({x}) = µ2({x}) = 1 or equivalently µ1 = µ2 = δx. Thus, δx isextreme and the map

δ : X → ∂eM+1 (X), x 7→ δx

18


is well defined.Let (xi)i∈I ⊆ X be a net that converges to some point x ∈ X. For every f ∈ C(X;R), we have

δxi(f) = f(xi)→ f(x) = δx(f),

so the net (δxi)i∈I converges to δx. This means that δ is continuous. Clearly δ is injective.Since ∂eM+

1 (X) is Hausdorff and X is compact, the only thing left to show is the surjectivity.Let µ ∈ ∂eM+

1 (X). Assume that there exists a Borel set C such that µ(C) ∈ (0, 1). Then theformulae

µ1(B) := µ(C)−1µ(B ∩ C) and µ2(B) := µ(X \ C)−1µ(B ∩ (X \ C)),

for any Borel set B, define elements µ1, µ2 in M+1 (X). Moreover, µ = µ(C)µ1 + µ(X \ C)µ2.

It follows that µ1 = µ2 = µ, since µ is assumed to be an extreme point. But then we get1 = µ1(C) = µ2(C) = 0, which is a contradiction. Thus, µ only attains the values 0 and 1.Let Γ denote the set of all compact subsets of X that have measure 1. Given B1, B2 ∈ Γ, theirunion also lies in Γ, hence

1 = µ(B1 ∪B2) = µ(B1) + µ(B2)− µ(B1 ∩B2) = 2− µ(B1 ∩B2),

or equivalently µ(B1 ∩ B2) = 1. This shows that Γ is closed under finite intersections. As aconsequence, Γ has the finite intersection property. Since X is compact, the set C :=

⋂Γ is

nonempty. Notice that any compact set B ⊆ X \C cannot lie in Γ, hence µ(B) = 0. The innerregularity of µ implies µ(X \ C) = 0 or equivalently µ(C) = 1.Assume that C contains two distinct points x and y. By Urysohn’s lemma, there exists acontinuous function f : X → [0, 1] satisfying f(x) = 0 and f(y) = 1. Then C can be writtenas the union of the proper compact subsets f−1([0, 1

2 ]) and f−1([12 , 1]). Neither of them is

contained in Γ, so both have measure 0. But then C must have measure 0 as well which is acontradiction. Thus, C contains a unique element x. We have that µ({x}) = µ(C) = 1, whichshows that µ = δx, finishing the proof.

2.1 Partially ordered abelian groups with order unit

Another important class of examples for compact convex sets consists of state spaces of partiallyordered abelian groups with order unit. On the other hand, one can assign to each compactconvex set a certain partially ordered abelian group with order unit. This rough duality betweencompact convex sets and partially ordered abelian groups with order unit is the key ingredientthat allows us to define a tensor product for compact convex sets later on.

19


2.3 Definition A partially ordered abelian group is an abelian group G, equipped with acompatible partial order. A OAG-morphism between partially ordered abelian groups G andH is an order-preserving group homomorphism G → H. Note that a group homomorphismG → H is order-preserving if and only if it is positive, i.e. maps positive elements to positiveelements. Let OAG be the category whose objects are ordered abelian groups and whosemorphisms are OAG-morphisms. We use G+ to denote the set of all positive elements in G.An element u ∈ G+ is called order unit if every element in G is dominated by some multipleof u, that is for every g ∈ G, there exists n ∈ N such that g ≤ nu.A partially ordered abelian group with order unit is a pair (G, u), consisting of a partiallyordered abelian group G and an order unit u in G. A GOU-morphism between partiallyordered abelian groups with order unit (G, u) and (H, v) is a OAG-morphism G → H thatmaps u to v. The category GOU shall consist of all partially ordered abelian groups with orderunit and all GOU-morphisms. We denote the set of all GOU-morphisms (G, u) → (R, 1) byS(G, u). The elements of S(G, u) are called states.We may shorten ‘partially ordered abelian group’ to ‘ordered abelian group’. Also, we mayabbreviate ‘partially ordered abelian group with order unit’ to ‘group with order unit’.

Remark Observe that an ordered abelian group G is directed if and only if G = G+ − G+.Every group with order unit is directed.

It is not trivial that a general group with order unit (G, u) has a state. However, as long as(G, u) is nonzero, this turns out to be true. This is a consequence of the following Proposition,the proof of which requires Zorn’s Lemma.

2.4 Proposition Let H be a subgroup of a partially ordered abelian group G such that eachelement in G is dominated by an element in H. Then every positive homomorphism H → Rextends to a positive homomorphism G→ R.

Proof. [Goo86, 4.2].

2.5 Corollary Let (G, u) be a partially ordered abelian group with order unit and supposethat H is a subgroup containing u. Then every state on (H,u) extends to a state on (G, u).

2.6 Corollary Let (G, u) be a partially ordered abelian group with order unit. There existsa state on (G, u) if and only if G is nonzero.


If (G, u) is a group with order unit, we can recover all OAG-morphisms G→ R from the states

20


on (G, u), as follows.

2.7 Lemma For any partially ordered abelian group with order unit (G, u), we have that

HomOAG(G,R) = ((0,∞) · S(G, u)) ∪ {0}.


2.8 Proposition Let (G, u) be a partially ordered abelian group with order unit. Then S(G, u)

is a compact convex subset of the locally convex, Hausdorff, real topological vector space RG

of all real-valued functions on G.Given two partially ordered abelian groups with order unit (G, u) and (H, v) and a morphismϕ : (G, u)→ (H, v), the map

ϕ∗ : S(H, v)→ S(G, u), ϕ∗(f) := f ◦ ϕ

is continuous and affine. Thus, the assignment (G, u) 7→ S(G, u) defines a contravariant functorfrom GOU to the category of compact convex sets (with continuous affine maps as morphisms).

Proof. (cf. [Goo86, 6.2]) It is evident that S(G, u) is a closed, convex subset of RG (closednesscan easily be checked via nets). For each x ∈ G, choose a natural number nx such that−nxu ≤ x ≤ nxu. It follows that −nx ≤ s(x) ≤ nx holds for each state s ∈ S(G, u), whence

S(G, u) ⊆∏x∈G

[−nx, nx].

The product of intervals is compact by Tychonoff’s theorem. It follows that S(G, u) is acompact convex set.The second part of the statement is just a routine verification and will be omitted.

2.9 Lemma Let X be a compact Hausdorff space. Then (C(X;R), 1) is a partially orderedabelian group with order unit. Moreover, S(C(X;R), 1) and M+

1 (X) are naturally isomorphicas compact convex sets.

Proof. (cf. [Goo86, 6.8]) It is clear that C(X;R) is a partially ordered abelian group. Since Xis compact, every function in C(X;R) is bounded, implying that 1 is a order unit.The inclusion C(X;R)∗ ↪→ RC(X;R) is an embedding of topological vector spaces, since bothcarry the topology of pointwise convergence. Thus, we may regard both S(C(X;R), 1) andM+

1 (X) as compact convex subsets of RC(X;R). The inclusionM+1 (X) ⊆ S(C(X;R), 1) is trivial.

For the converse inclusion, let ϕ ∈ S(C(X;R), 1). We have to show that ϕ is homogeneous. Fort ∈ R and f ∈ C(X;R), let us prove that the formula ϕ(tf) = tϕ(f) holds. We may assume

21


without loss of generality that f is positive. It is clear that the formula holds if t is rational.In the general case, let α, β ∈ Q such that α ≤ t ≤ β. Then αf ≤ tf ≤ βf , which implies that

αϕ(f) = ϕ(αf) ≤ ϕ(tf) ≤ ϕ(βf) = βϕ(f).

Since α and β may be arbitrarily close to t, we obtain ϕ(tf) = tϕ(f), as desired.

2.10 Proposition Let K be a compact convex set. Then (Aff(K), 1) is a partially orderedabelian group with order unit.Given two compact convex sets K1 and K2 and a continuous affine map f : K1 → K2, the map

f∗ : (Aff(K2), 1)→ (Aff(K1), 1), f∗(g) := g ◦ f

is a GOU-morphism. Thus, the assignment K 7→ (Aff(K), 1) defines a contravariant functorfrom the category of compact convex sets to GOU.

Proof. Using the compactness of K, we deduce that every continuous map K → R is boundedfrom above. It follows that 1 is an order unit for Aff(K). The rest of the proof consists oftrivial verifications and will be omitted.

We have seen so far that there are contravariant functors

Compact convex sets � GOUK 7−→ (Aff(K), 1)

S(G, u) ←− [ (G, u)

For any partially ordered abelian group with order unit (G, u), the map

ι : (G, u)→ (Aff(S(G, u)), 1), g 7→ evg

is a GOU-morphism. In fact, ι is a natural transformation from the identity functor to thefunctor (Aff(S(_)), 1). Thus, we will refer to ι as the natural map. In general, ι will not bean isomorphism. This is not surprising as (G, u) may lack characteristics that (Aff(S(G, u)), 1)

always has, such as being divisible and unperforated. For example (Aff(S(Z, 1)), 1) ∼= (R, 1),which is not isomorphic to (Z, 1). It is known that ι is an embedding if and only if G isarchimedean (see [Goo86, 7.7]).Similarly, if K is a compact convex set, the map

κ : K → S(Aff(K), 1), x 7→ evx

is continuous and affine. Again, κ is a natural transformation from the identity functor to thefunctor S(Aff(_)), so we refer to κ as the natural map. Contrary to ι, this map is always anisomorphism, as shown in the following proposition.

22


2.11 Proposition For any compact convex set K, the natural map κ : K → S(Aff(K), 1) isan isomorphism.

Proof. (cf. [Goo86, 7.1]) Suppose that K is located in a locally convex, Hausdorff, real topo-logical vector space E. For the injectivity, let x, y ∈ K such that x 6= y. By the Hahn-Banachtheorem, there exists a continuous linear map f : E → R such that f(x) 6= f(y). The restrictionof f to K is continuous and affine, and we have that

κ(x)(f |K) = f(x) 6= f(y) = κ(y)(f |K),

hence κ(x) 6= κ(y).To show that κ is surjective, let s ∈ S(Aff(K), 1). For now, let us assume that s is an extremepoint. Clearly, the restriction map

ρ : M+1 (K) = S(C(K;R), 1)→ S(Aff(K), 1)

is continuous and affine. Moreover, it is surjective according to Corollary 2.5. Notice thatρ−1({s}) is a nonempty compact face of M+

1 (K). By the Krein-Milman theorem, there existsat least one extreme point δ ∈ ∂eρ−1({s}). It follows that δ ∈ ∂eM+

1 (K), hence by Proposition2.2, there exists an element x ∈ K such that δ = δx. For all f ∈ Aff(K), we get

s(f) = ρ(δ)(f) = δx(f) = f(x) = κ(x)(f),

which implies that s = κ(x). So far, we have shown that ∂eS(Aff(K), 1) is contained in theimage of κ. Note that the latter is a compact convex set, so applying the Krein-Milman theoremonce again yields

S(Aff(K), 1) = conv(∂eS(Aff(K), 1)) ⊆ κ(K),

which just means that κ is in fact surjective. As a continuous bijection from a compact spaceto a Hausdorff space, κ is a homeomorphism.

It follows from Proposition 2.11 that every compact convex set is isomorphic to a state spaceof some partially ordered abelian group with order unit.

2.2 Approximations of (semi-)continuous functions

One of our highest priorities right now is to show that LAff(K)0++ is a Cu-semigroup. The

difficulty lies in showing that every element in LAff(K)0++ may be approximated by elements

that are way-below. The key observation is that every lower semicontinuous, affine functionmay be approximated by an increasing net of continuous, affine functions (see Proposition

23


2.15). Throughout this section, we will prove various other approximation results which areneeded in Chapter 3.

Let X be a compact Hausdorff space. We use Lsc(X) to denote the set of all lower semicon-tinuous functions X → (−∞,∞]. The sets Lsc(X)+ and Lsc(X)0

++ are defined analogously tothe case of (lower semi-)continuous, affine functions.

2.12 Lemma 1) Let X be a nonempty, compact Hausdorff space. Then any f ∈ Lsc(X)

attains its minimum.2) Let K be a nonempty, compact convex set. Then any f ∈ LAff(K) attains its minimum on

the extreme boundary of K.

Proof. 1) For any n ∈ N, the set f−1((−n,∞]) is open in X by the lower semicontinuityof f . Since X =

⋃n∈N f

−1((−n,∞]) is compact, there exists a number n ∈ N such thatX = f−1((−n,∞]). Therefore f is bounded below, so α := infx∈X f(x) is a real number.Choose a net (xλ)λ∈Λ ⊆ X such that α = limλ f(xλ). Since X is compact, this net has aconvergent subnet. In order to keep notation simple, we may assume without loss of generalitythat the net (xλ)λ∈Λ is already converging to some z ∈ X. Using the lower semicontinuity off at the third step, we deduce

α ≤ f(z) = f(limλxλ) ≤ lim

λf(xλ) = α,

hence f attains its minimum at the point z.2) Let α denote the infimum of f and consider F := {x ∈ K | f(x) = α} = f−1((−∞, α]).Since f is lower semicontinuous and affine, F is a compact convex set which is nonempty bythe previous part. It is easily seen that F is a face of K. By the Krein-Milman theorem, F hasat least one extreme point z. Then z is also an extreme point of K satisfying f(z) = α.

That every lower semicontinuous, affine function can be approximated by continuous, affinefunctions heavily relies on the following application of the Hahn-Banach theorem.

2.13 Lemma Let E be a locally convex, Hausdorff, real topological vector space, and letA,B ⊆ E ⊕ R be closed, disjoint convex subsets such that A is compact. Assume that thereexist z ∈ E and µ1, µ2 ∈ R such that (z, µ1) ∈ A, (z, µ2) ∈ B and µ1 < µ2. Then there existsa continuous, affine function e : E → R such that

A ⊆ {(y, t) ∈ E ⊕ R | t < e(y)} and B ⊆ {(y, t) ∈ E ⊕ R | t > e(y)}.

Proof. By the Hahn-Banach separation theorem, there exists a continuous linear functionalϕ ∈ (E ⊕ R)∗ such that

sup(y,t)∈A

ϕ(y, t) < inf(y,t)∈B

ϕ(y, t).

24


Let ψ := ϕ(_, 0), γ := ϕ(0, 1). For every point (y, t) ∈ E⊕R, we have ϕ(y, t) = ψ(y) + γ · t bythe linearity of ϕ. The formula above may be rewritten as

sup(y,t)∈A

ψ(y) + γ · t < inf(y,t)∈B

ψ(y) + γ · t.

The assumptions (z, µ1) ∈ A and (z, µ2) ∈ B entail ψ(z) + γ · µ1 < ψ(z) + γ · µ2, and theassumption µ1 < µ2 then implies that γ > 0. Choose any real number η such that

sup(y,t)∈A

ψ(y) + γ · t < η < inf(y,t)∈B

ψ(y) + γ · t.

Now define e : E → K via the formula e(x) := γ−1 · (η − ψ(x)). As ψ is continuous and linear,e is continuous and affine. Given (x, λ) ∈ A, the inequality ψ(x) + γ · λ < η holds. This isequivalent to the inequality λ < e(x), proving that A ⊆ {(y, t) ∈ E ⊕R | t < e(y)}. The otherclaimed inclusion can be shown similarly.

Let K be a compact convex set in some surrounding space E. The epigraph of a functionf : K → [−∞,∞] is defined as the set M := {(x, λ) ∈ K × R | f(x) ≤ λ}. Notice that M is aconvex subset of E ⊕ R if and only if f is convex. Also, M is a closed subset of E ⊕ R if andonly if f is lower semicontinuous.

2.14 Lemma ([Alf71, I.1.2]) Let K be a compact convex set, and let f : K → (−∞,∞] be alower semicontinuous, convex function. For every x ∈ K, we have that

f(x) = sup{e(x) | e ∈ Aff(K), e < f}.

Proof. Let x ∈ K and α < f(x) be arbitrary. We have to show that there exists a continuous,affine function e ∈ Aff(K) such that e < f and α < e(x).Let us denote the epigraph of f by M . We would like to apply Lemma 2.13 to separate thepoint (x, α) fromM by means of a continuous affine function. However,M does not contain anypoint of the form (x, λ) if f(x) =∞, which prevents us from doing so. In order to circumventthis inconvenience, we will instead replace M by a bigger set M ′. Choose any real numberβ satisfying α < β < f(x) and set M ′ := conv({(x, β)} ∪M). Obviously M ′ is a nonemptyclosed convex subset of E ⊕ R. To show that (x, α) is not contained in M ′, we define thefollowing sets:

A := {(z, λ) ∈ K × R | β ≤ λ}, B := {(z, λ) ∈ K × R | f(z) ≤ λ ≤ β}.

Observe that conv({(x, β)}∪M) ⊆ A∪ conv({(x, β)}∪B). Notice that B is a compact convexset because f is lower semicontinuous and convex. It follows that conv({(x, β)} ∪B) is closedand consequently that

M ′ ⊆ A ∪ conv({(x, β)} ∪B).

25


Clearly, (x, α) is not contained in the right hand side, so it is not contained in M ′ either.We can now apply Lemma 2.13 to obtain a continuous, affine function e ∈ Aff(K) satisfyingα < e(x) and M ′ ⊆ {(z, λ) ∈ K × R | λ > e(z)}. To show that e < f , let y ∈ K be arbitrary.If f(y) = ∞, then e(y) < f(y) is trivially satisfied. In the case that f(y) < ∞, the element(y, f(y)) lies in M ′, implying that e(y) < f(y). This finishes the proof.

2.15 Proposition ([Alf71, I.1.4]) Let K be a compact convex set. For any function f inLAff(K), the set Λ := {e ∈ Aff(K) | e < f} is upward directed and has supremum f .

Proof. The statement is clear if f =∞, so let us assume that f 6=∞. We know from Lemma2.12 that f is bounded below, so Λ is nonempty. The epigraph M of f is a nonempty closedconvex subset of E ⊕ R. Let e1, e2 ∈ Λ and choose a real number η such that η ≤ e1, e2. Fori = 1, 2 consider the sets Mi := {(z, λ) ∈ K ×R | η ≤ λ ≤ ei(x)}. It follows from the fact thate1, e2 < f that M and M1 ∪M2 are disjoint. In fact, it it true that M ∩ conv(M1 ∪M2) = ∅,since f is affine. Clearly, each Mi is a nonempty compact convex set, so conv(M1 ∪M2) is alsoa nonempty compact convex set. We can now apply Lemma 2.13 to obtain a continuous affinefunction e ∈ Aff(K) such that

conv(M1 ∪M2) ⊆ {(z, λ) ∈ K × R | λ < e(z)} and M ⊆ {(z, λ) ∈ K × R | λ > e(z)}.

Similarly as in the proof of Lemma 2.14, this will imply that e1, e2 < e < f , proving that Λ isin fact an upward directed set. It follows from Lemma 2.14 that is has supremum f .

2.16 Lemma Let K be a compact convex set.1) Let f : K → [−∞,∞) be upper semicontinuous and affine, g ∈ LAff(K). Then f ≤ g if and

only if f |∂eK ≤ g|∂eK . Moreover, f < g if and only if f |∂eK < g|∂eK if and only if thereexists an ε > 0 such that f + ε ≤ g.

2) Let f , g ∈ LAff(K). Then f ≤ g if and only if f |∂eK ≤ g|∂eK .

Proof. We only show the nontrivial implications.1) The function g− f is lower semicontinuous and affine, so it attains its minimum on ∂eK byLemma 2.12. Let ε be the minimum, then f + ε ≤ g. The assumption f |∂eK ≤ g|∂eK impliesε ≥ 0, while the assumption f |∂eK < g|∂eK entails ε > 0.2) By Proposition 2.15, there exists an upward directed set Λ ⊆ Aff(K) with supremum f . Foreach e ∈ Λ, we have that e|∂eK ≤ f |∂eK ≤ g|∂eK , hence e ≤ g by the first part. By passing tothe supremum, we obtain f ≤ g.

2.17 Lemma Let K be a compact convex set and suppose that f, g : K → R are lowersemicontinuous, convex and bounded above (for example, f and g are continuous). There

26


exists a biggest convex function e less than f and g. Moreover, e is lower semicontinuous.

Proof. (cf. [Goo86, 11.10]) Suppose that K is located in some locally convex, Hausdorff, realtopological vector space E. Choose real numbers α and β such that α ≤ f, g ≤ β. The sets

Mf := {(x, λ) ∈ K × R | f(x) ≤ λ ≤ β} and Mg := {(x, λ) ∈ K × R | g(x) ≤ λ ≤ β}

are compact, convex subsets of E⊕R since both f and g are assumed to be lower semicontinuousand convex. The setMe := conv(Mf∪Mg) is convex, but also compact, becauseMf andMg areboth convex and compact. Note thatMe is contained in the set {(x, λ) ∈ K×R | α ≤ λ}, sincethe latter is a convex set containingMf andMg. It follows that e : K → R, e(x) := inf(x,λ)∈Me

λ

is a well defined function. By the compactness of Me, we may change the infimum to aminimum. We prove that e is convex: Let x, y ∈ K, t ∈ [0, 1]. Both the points (x, e(x)) and(y, e(y)) lie in Me. As Me is convex, the point (tx+ (1− t)y, te(x) + (1− t)e(y)) must also liein Me, implying that e(tx+ (1− t)y) ≤ te(x) + (1− t)e(y).If x lies in K, then the point (x, f(x)) lies in Mf ⊆ Me, implying that e(x) ≤ f(x). As x isarbitrary, we conclude that e ≤ f . Similarly, we obtain that e ≤ g. Now suppose that h isanother convex function such that h ≤ f, g. The set Mh := {(x, λ) ∈ K × R | h(x) ≤ λ ≤ β}is again convex by the convexity of h. The inequality h ≤ f, g implies that Mf ,Mg ⊆Mh. Weobtain that Me ⊆Mh, which again implies that h ≤ e. Thus, e is the biggest convex functionless than f and g.Before we move on to the last part of this proof, we claim that

Me = {(x, λ) ∈ K × R | e(x) ≤ λ ≤ β}.

If (y, µ) lies in Me, then e(y) ≤ µ holds by definition. Just like in the beginning of the proof,Me is contained in the set {(x, λ) ∈ K × R | λ ≤ β}, so in particular µ ≤ β. Conversely,suppose that (y, µ) ∈ K ×R satisfies e(y) ≤ µ ≤ β. The set A := {λ | (y, λ) ∈Me} is a convexsubset of R (and in particular connected). Clearly, the points e(y) and β lie in A, implyingthat [e(y), β] ⊆ A. Thus, µ lies in A, or in other words (y, µ) ∈Me.Finally, let us show that e is lower semicontinuous. Given t ∈ R, we will prove that e−1((−∞, t])is closed. This is clear if β < t, since e−1((−∞, t]) = K in this case. Now let us assume thatt ≤ β. The function ϕ : K → E ⊕ R, ϕ(x) := (x, t) is continuous. It follows that

e−1((−∞, t]) = {x ∈ K | e(x) ≤ t} = {x ∈ K | e(x) ≤ t ≤ β}= {x ∈ K | (x, t) ∈Me} = ϕ−1(Me)

is a closed subset of K, as desired.

27


2.18 Lemma Let K be compact convex set, and let f : K → [−∞,∞) be an upper semicon-tinuous, convex function. For every x ∈ K, we have that

f(x) = inf{e(x) | e : K → R continuous, convex, f < e}.

Proof. (cf. [Goo86, 11.9]) Let x ∈ K. As f is upper continuous, it has an upper bound whichwe denote by γ. Let α be an arbitrary real number such that f(x) < α. We have to show thatthere exists a continuous, convex function e : K → R such that f < e and e(x) < α. If α > γ,we can choose e to be the constant function with value 2−1(γ + α). So let us assume thatα ≤ γ. Choose a real number β such that f(x) < β < α. Since f is upper continuous, the setf−1([β,∞)) is compact and does not contain x. Choose an open neighborhood U of x in thesurrounding locally convex space E such that U and f−1([β,∞)) are disjoint (this is a smallargument involving compactness of f−1([β,∞))). We can write U as U = x + V , where V isan open neighborhood of the origin in E. Since E is a locally convex space, we may assumewithout loss of generality that V is balanced (meaning that λV ⊆ V for every λ ∈ [0, 1]), convexand open. Let p denote the Minkowski functional associated with V . Then p is the uniqueseminorm on E satisfying V = {y ∈ E | p(y) < 1}. It is also true that V = {y ∈ E | p(y) ≤ 1}.Define e : K → R by the formula

e(y) := β + (γ − β)p(y − x).

Then e is continuous by the continuity of p. Also, since γ − β > 0 and since p is convex, e isconvex as well. Moreover, e(x) = β < α. It is left to show that f < e. Given y ∈ K, let usshow that f(y) < e(y). Note that β ≤ e, so we may assume without loss of generality thatβ ≤ f(y). In this case, y is contained in f−1([β,∞)), which implies that it is not contained inU = x+ V , therefore p(y − x) > 1. It follows that

e(y) = β + (γ − β)p(y − x) > β + (γ − β) · 1 = γ ≥ f(y),

as desired.

2.19 Lemma Let X be a compact Hausdorff space, f : X → [−∞,∞) upper semicontinuousand Λ ⊆ Lsc(X) nonempty such that f < sup Λ. Then there exist finitely many elementse1, . . . , en ∈ Λ such that f < max{e1, . . . , en}. If Λ is upward directed, there exists an elemente ∈ Λ such that f < e.

Proof. For any x ∈ X, we have f(x) < supe∈Λ e(x). Choose ex ∈ Λ such that f(x) < ex(x).Since ex − f is lower semicontinuous, the set M(x) := {y ∈ X | 0 < ex(y) − f(y)} is an openneighborhood of x. By compactness of X =

⋃x∈XM(x), there exist x1, . . . , xn ∈ X such that

K = M(x1)∪ . . .∪M(xn), implying that f < max{ex1 , . . . , exn}. If Λ is assumed to be upwarddirected, we can choose an element e ∈ Λ that dominates each exi which entails f < e.

28


2.20 Lemma Let K be a compact convex set, f : K → [−∞,∞) upper semicontinuous andconvex and h : K → (−∞,∞] lower semicontinuous and concave such that f < h. Then thereexists a lower semicontinuous and convex function e : K → R such that f < e < h.

Proof. We know from Lemma 2.18 that the set

Λ := {e | e : K → R continuous, convex, f < e}

has f as its infimum, hence inf Λ < h. By Lemma 2.19 (or rather its dual result), there existfinitely many elements e1, . . . , en ∈ Λ such that min{e1, . . . , en} < h. Choose a positive realnumber ε > 0 such that f + ε < e1, . . . , en. By Proposition 2.17, there exists a biggest convexfunction e : K → R less than e1, . . . , en, and e is moreover lower semicontinuous. It followsthat f < f + ε ≤ e ≤ min{e1, . . . , en} < h.

2.21 Lemma Let K be a compact convex set, f : K → [−∞,∞) upper semicontinuous andconvex and h : K → (−∞,∞] lower semicontinuous and concave such that f < h. There existsa function f ′ : K → R that is the pointwise maximum of finitely many functions from Aff(K)

and such that f < f ′ < h.

Proof. By Lemma 2.20, there exists a lower semicontinuous, convex function e : K → R suchthat f < e < h. We have seen in Lemma 2.14 that the set

Λ := {e′ ∈ Aff(K) | e′ < e}

has supremum e. Using Lemma 2.19, there exist finitely many elements e1, . . . , en ∈ Λ suchthat f < max{e1, . . . , en}. The function f ′ := max{e1, . . . , en} has the desired properties.

2.22 Proposition Let K be a compact convex set, f : K → [−∞,∞) upper semicontinuousand convex and h : K → (−∞,∞] lower semicontinuous and concave such that f < h. Thereexists a function f ′ : K → R that is the pointwise maximum of finitely many functions fromAff(K) and a function h′ : K → R that is the pointwise minimum of finitely many functionsfrom Aff(K) such that f < f ′ < h′ < h.

Proof. (cf. [Goo86, 11.11]) The existence of such an f ′ is given by Lemma 2.21. Then −his upper semicontinuous and convex, while −f ′ is continuous and concave with −h < −f ′.Applying Lemma 2.21 a second time, we get a function e that is the pointwise maximumof finitely many functions from Aff(K) and such that −h < e < −f ′. Then h′ := −e is afunction that is the pointwise minimum of finitely many functions from Aff(K). Moreover,f < f ′ < h′ < h.

29


2.3 LAff(K)0++ is a Cu-semigroup

Now that we know that every lower semicontinuous, affine function is the supremum of anincreasing net of continuous, affine functions, we can show that LAff(K)0

++ is a Cu-semigroup.However, in order to pass from nets to sequences, we have to assume that K is metrizable.

2.23 Lemma Let X be a compact Hausdorff space. For any system B ⊆ O(X) of opensubsets of X, the following are equivalent:1) B is a basis for the topology on X.2) B separates the points of X, that is: for all x, y ∈ X, x 6= y, there exist U, V ∈ B such that

x ∈ U , y ∈ V and U ∩V = ∅. Also, for all x ∈ X and U1, . . . , Un ∈ B such that x ∈ ∩ni=1Ui,there exists U ∈ B such that x ∈ U ⊆ ∩ni=1Ui.

Proof. 1) ⇒ 2): Let x, y ∈ X such that x 6= y. Choose open, disjoint neighborhoods A of xand B of y. Since B is assumed to be a basis for the topology on X, we can find U, V ∈ B suchthat x ∈ U ⊆ A and y ∈ V ⊆ B. Clearly U and V are disjoint, so this shows that B separatesthe points of X. The second condition in 2) is trivially satisfied.2) ⇒ 1): Let x ∈ X, and let A be an open neighborhood for x. For each y ∈ X \ A, we maychoose disjoint sets Uy, Vy ∈ B such that x ∈ Uy and y ∈ Vy. Since X \ A is compact andcontained in ∪y∈X\UVy, we find y1, . . . , yn ∈ X \A such that X \A ⊆ ∪ny=1Vyi . It follows that

x ∈ ∩ni=1Uyi ⊆ ∩ni=1X \ Vyi ⊆ A.

Finally, we may choose U ∈ B such that x ∈ U ⊆ ∩ni=1Uyi ⊆ A, finishing the proof.

2.24 Proposition Let K be a compact convex set. Then LAff(K) is a domain. For functionsf, h ∈ LAff(K), we have f � h if and only if there exist g ∈ Aff(K) such that f ≤ g < h.Additionally, the following conditions are equivalent:1) K is metrizable.2) Aff(K) is separable.3) LAff(K) is countably based.

Proof. Let f, h ∈ LAff(K). Suppose that f � h. The set Λ := { e | e ∈ Aff(K), e < h} isupward directed with supremum h according to Proposition 2.15, hence there exists an elementg ∈ Λ such that f ≤ g. Conversely, suppose there exist g ∈ Aff(K) such that f ≤ g < h. Toshow that f � h, let Λ ⊆ LAff(K) be an upward directed set with h ≤ sup Λ. By Lemma2.19, there exists an element e ∈ Λ such that f ≤ g < e.1) ⇒ 2): If we assume that K is metrizable, then C(K;R) is separable. Since any subspace ofa separable metrizable space is separable itself, we obtain that Aff(K) is separable.

30


2) ⇒ 3): Suppose that Aff(K) is separable, and let B ⊂ Aff(K) be a countable dense subset.We claim that B is a base for LAff(K). Let f, h ∈ LAff(K) such that f � h. By the descriptionof the way-below relation, there exists a function g ∈ Aff(K) such that f ≤ g < h. Chooseε > 0 such that f ≤ g < g+ 2ε ≤ h, then choose an element b ∈ B such that ‖b− (g+ ε)‖ < ε.It follows that −ε < b− (g + ε) < ε, hence f ≤ g < b < g + 2ε ≤ h, as desired.3) ⇒ 1): Let B ⊆ LAff(K) be a countable basis. We claim that the system

A := {f−1((0,∞]) | f ∈ B} ⊆ O(K)

separates the points ofK. To prove this claim, let x, y ∈ K such that x 6= y. Choose h ∈ Aff(K)

such that h(x) 6= h(y). We can arrange that h(x) = 2 while also h(y) = −2. Choose f, g ∈ Bsatisfying h− 1 ≤ f < h and −h− 1 ≤ g < −h. We find that

f(x) ≥ h(x)− 1 = 1, i.e. x ∈ f−1((0,∞]),

g(y) ≥ −h(y)− 1 = 1, i.e. y ∈ g−1((0,∞]),

and that f−1((0,∞]) ∩ g−1((0,∞]) = ∅: if z lies in f−1((0,∞]), then h(z) > f(z) > 0, whichimplies that g(z) < −h(z) < 0, hence z /∈ g−1((0,∞]). This proves the claim that A separatesthe points of K. Now let B be the set of all finite intersections of elements in A. Then B iscountable, separates the points of K and is trivially closed under finite intersections. It followsfrom Lemma 2.23 that B is a basis for the topology on K. Therefore K is second countableand consequently metrizable.

2.25 Proposition Let K be a metrizable compact convex set. Then LAff(K)0++ is a Cu-

semigroup. It is countably based, simple and satisfies (O5). Given f, h ∈ LAff(K)0++ with

h 6= 0, we have f � h in LAff(K)0++ if and only if f � h in LAff(K).

Proof. Most of the statements follow easily from Proposition 2.24, and it is clear that LAff(K)0++

is simple. In order to prove that LAff(K)0++ satisfies (O5), let f, f ′, g, g′, h ∈ LAff(K)0

++ suchthat

f ′ � f, and g′ � g, and f + g ≤ h.

Choose a continuous affine function f ′′ such that f ′ ≤ f ′′ � f . One can then show that thefunction e := h− f ′′ lies in LAff(K)0

++ and satisfies f ′ + e ≤ h ≤ f + e and g′ ≤ e.

31

3 Choquet simplices

3.1 Definition and examples

Let G be an abelian group. A cone in G is just another term for a submonoid in G. Any coneC in G defines a reflexive, transitive order on G by setting

x ≤C y :⇐⇒ y − x ∈ C,

for x, y ∈ C. Notice that C equals the set of all positive elements in (G,≤C), and that theorder is compatible with addition. This order is a partial order if and only if C is strict, thatis if C ∩ (−C) = {0}, or equivalently, if C is a conical monoid, that is if the equality x+ y = 0

implies that x = y = 0, for x, y ∈ C.

3.1 Definition A convex cone in some locally convex, Hausdorff, real topological vector spaceE is a convex subset C that is also a cone in the abelian group (E,+). Notice that any convexcone is closed under multiplication with positive scalars. We call C strict if the underlying coneis strict. A lattice cone is a strict convex cone C such that (C,≤C) is a lattice, or equivalently,such that (C − C,≤C) is a lattice.A map C → D between two convex cones is called linear if it is affine and preserves addition.Such a map will automatically be homogeneous for positive scalars.

3.2 Proposition Let K be a compact convex set. There exists a strict convex cone C(K)

and a continuous affine map κ : K → C(K) such that the following universal property holds:For every convex cone D and every affine map ϕ : K → D, there exists a unique linear mapϕ : C(K) → D such that ϕ = ϕ ◦ κ. Moreover, ϕ is continuous (lower semicontinuous, uppersemicontinuous) if and only if ϕ is continuous (lower semicontinuous, upper semicontinuous).Furthermore, C(K) is unique up to linear homeomorphism and is referred to as the cone withbase K.

Proof. Let κ : K → S(Aff(K), 1) be the natural isomorphism. Then C(K) can be realized as

C(K) := HomOAG(Aff(K),R) = {λ · κ(x) | λ > 0, x ∈ K} ∪ {0},

32

3 Choquet simplices

where the equality follows from Lemma 2.7. Notice that C(K) is a strict convex cone, andthat κ : K → C(K) is continuous and affine.Assume that D is a convex cone and that ϕ : K → D is an affine map. We define ϕ : C(K)→ D

by ϕ(0) := 0 and ϕ(λ · κ(x)) := λ · ϕ(x), for λ > 0 and x ∈ K. The representation of anelement in C(K) \ {0} as λ · κ(x) is unambiguous, so ϕ is well defined. It is a straightforwardverification that ϕ has the desired properties and will be omitted. The statements concerning(semi-)continuity follow from the fact that the map (0,∞) · κ(K) → (0,∞) × K, given byλ · κ(x) 7→ (λ, x), is continuous.

Remark We say that K is regularly embedded in the surrounding space E if K lies in ahyperplane that misses the origin. If this is the case, the cone with base K can be realized asC(K) = ((0,∞) ·K) ∪ {0}. In fact, the proof of Proposition 3.2 relies on the fact that K isisomorphic to S(Aff(K), 1), and that the latter is regularly embedded in RAff(K).

3.3 Definition A Choquet simplex is a compact convex set K such that the cone with baseK is a lattice cone. It follows from Proposition 3.2 that this definition does not depend on theconcrete realization of the cone with base K.

Before we give examples of Choquet simplices, let us briefly discuss which compact convex setsare not Choquet simplices. It is shown in [Goo86, 10.8] that the extreme boundary of anyChoquet simplex K is affinely independent. It follows that squares, cubes, disks, balls and infact most compact convex sets are not Choquet simplices. With this in mind, one can showthat up to affine homeomorphism, the only Choquet simplices that can be embedded into finitedimensional spaces are the empty set and the standard n-dimensional simplices, for n ∈ N. Thestandard n-simplex is defined as the convex hull of the (n+ 1) standard basis vectors in Rn+1.For more interesting examples of Choquet simplices, one has to consider infinite dimensionalcompact convex sets.

3.4 Definition We say that a partially ordered set (X,≤) has the Riesz interpolation property(or just interpolation) if for all x1, x2, z1, z2 ∈ X satisfying x1, x2 ≤ z1, z2, there exists y ∈ Xsuch that x1, x2 ≤ y ≤ z1, z2.An abelian semigroupM is said to have the Riesz refinement property if for all x1, x2, z1, z2 ∈Mwith x1 + x2 = z1 + z2, there exist yi,j ∈ M , for i, j ∈ {1, 2}, such that xi = yi,1 + yi,2 andzj = y1,j + y2,j , for i, j ∈ {1, 2}.A partially ordered abelian semigroup (M,≤) has the Riesz decomposition property if for allx, z1, z2 ∈ M satisfying x ≤ z1, z2, there exist y1, y2 ∈ M such that y1 ≤ z1, y2 ≤ z2 andx = y1 + y2.

It is well known that an ordered abelian group (G,≤) has the Riesz interpolation property

33

3 Choquet simplices

if and only if G+ has the Riesz refinement property if and only if (G+,≤) has the Rieszdecomposition property. A proof can be found in [Goo86, 2.1]. An ordered abelian group withthe Riesz interpolation property is also called interpolation group.

3.5 Proposition Let (G, u) be a partially ordered group with order unit. If G has interpola-tion, then S(G, u) is a Choquet simplex.

Proof. As S(G, u) is regularly embedded in RG, the cone with base S(G, u) can be realized as

C = ((0,∞) · S(G, u)) ∪ {0} = HomOAG(G,R).

The last equality is given by Lemma 2.7. Let f, g ∈ C and definem : G+ → R+ as the pointwisemaximum of f and g.The following vocubulary refers solely to this proof: a decomposition of an element x ∈ G+ isa finite collection of elements x1, . . . , xn ∈ G+ such that x1 + . . .+ xn = x.Let x ∈ G+. For every decomposition x = x1 + . . .+ xn, we have that

m(x1) + . . .+m(xn) ≤ (f + g)(x1) + . . .+ (f + g)(xn) = (f + g)(x),

hence the set

M(x) := {m(x1) + . . .+m(xn) | x1, . . . , xn is a decomposition of x}

is bounded above by (f + g)(x). Therefore the function h : G+ → R+, h(x) := supM(x) iswell defined. We claim that the equation h(x + y) = h(x) + h(y) holds for all x, y ∈ G+. Forx, y ∈ G+, let

x = x1 + . . .+ xn and y = y1 + . . .+ ym

be arbitrary decompositions. Then x1 + . . .+ xn + y1 + . . .+ ym is a decomposition of x+ y,implying that

m(x1) + . . .+m(xn) +m(y1) + . . .+m(ym) ≤ h(x+ y).

This means that tx+ ty ≤ h(x+ y) whenever tx ∈M(x) and ty ∈M(y), from which we deduce

h(x) + h(y) = suptx∈M(x)

supty∈M(y)

tx + ty ≤ h(x+ y).

The converse inequality is where the interpolation comes into play. Let x+y = z1 + . . .+zn bea decomposition. Since G has interpolation, or equivalently, since G+ has the Riesz refinementproperty, there exist x1, . . . , xn, y1, . . . , yn ∈ G+ such that

x = x1 + . . .+ xn,

y = y1 + . . .+ yn and

zi = xi + yi for each i ∈ {1, . . . , n}.

34

3 Choquet simplices

For each i ∈ {1, . . . , n}, we have that m(zi) ≤ m(xi) +m(yi). This implies that

m(z1) + . . .+m(zn) ≤ m(x1) + . . .+m(xn) +m(y1) + . . .+m(yn) ≤ h(x) + h(y),

which yields h(x+ y) ≤ h(x) + h(y) after passing to the supremum.We have shown that h(x+y) = h(x)+h(y) if x, y ∈ G+. Note that h(0) = 0. As G is directed,h extends uniquely to an element h ∈ C. We claim that h is the supremum of f and g in C. Forevery x ∈ G+, we have f(x), g(x) ≤ m(x) ≤ h(x), which implies that f, g ≤C h. Thus, h is anupper bound for f and g. Let e ∈ C be another upper bound for f and g. For every x ∈ G+,we get m(x) = max{f(x), g(x)} ≤ e(x). Just as we did before, one can show that e(x) is anupper bound for M(x) which means that h(x) = supM(x) ≤ e(x). It follows that h ≤C e, soh is in fact the supremum of f and g in C. We have shown that (C,≤C) is a sup-semilattice.That (C,≤C) is an inf-semilattice can be shown similarly. It follows that S(G, u) is in fact aChoquet simplex.

3.6 Corollary If X is a compact Hausdorff space, then M+1 (X) is a Choquet simplex.

Proof. We have seen in Lemma 2.9 that M+1 (X) ∼= S(C(X;R), 1). Clearly C(X;R) has inter-

polation, so the assertion follows from the previous proposition.

3.2 Continuous and affine functions on Choquet simplices

Choquet simplices are interesting for us because their continuous and affine functions satisfythe Riesz interpolation property. The proof requires the following technical Lemma, which isadapted from [Goo86, 11.1 and 11.2].But first, let us settle some notation. Let X be a compact Hausdorff space. On C(X;R), wedefine a partial order <= by setting f <= g if and only if f = g or f < g. We refer to <= asthe strict ordering. For a linear subspace A ⊆ C(X;R) containing 1, observe that

HomOAG((A,<=),R) = HomOAG((A,≤),R) ⊆ A∗,

where A∗ denotes the topological dual space of A. That any ϕ ∈ HomOAG((A,≤),R) is linearcan be shown similarly as in the proof of Lemma 2.9. It follows just like in the proof ofProposition 2.1 that ϕ is bounded (the assumption that 1 ∈ A is used here). We use ≤+ todenote the order on A∗ induced by the cone HomOAG((A,<=),R).

3.7 Lemma Let X be a compact Hausdorff space, and let A ⊆ C(X;R) be a linear subspacecontaining 1. Equip A with the strict ordering. Suppose that (A∗,≤+) is a lattice, and letp ∈ A∗. The function r : A+ → R, defined by

r(f) := sup{p(e) | e ∈ A, 0 <= e <= f},

35

3 Choquet simplices

is additive.

Proof. Let p+ denote the supremum of p and 0 in (A∗,≤+), and let f ∈ A+. Given e ∈ A suchthat 0 <= e <= f , we get p(e) ≤ p+(e) ≤ p+(f). Thus, the set {p(e) | e ∈ A, 0 <= e <= f} isbounded above by p+(f). This shows that r is well defined and that r(f) ≤ p+(f).The converse inequality is clear if f = 0, so let us assume that f > 0. Let ε > 0, and choosen ∈ N∗ such that r(f)− p(f) ≤ nε. Consider the following subspaces of A⊕A:

C := {(e,−e) | e ∈ A} and D := R · (nf, f) + C.

Define g : D → R by g(γ · (nf, f) + (e,−e)) := γn(r(f) + ε) + p(e). Note that (nf, f) doesnot lie in C, hence the representation of the input is unique and g is well defined. Evidently, gpreserves the addition. We claim that g is positive. Let γ · (nf, f) + (e,−e) be positive, that is

0 <= γnf + e and 0 <= γf − e.

Adding these inequalities yields 0 <= γ(n+ 1)f , or equivalently 0 ≤ γ. The assumption γ = 0

results in e = 0, so g(γ · (nf, f) + (e,−e)) = 0. Let us therefore assume that γ > 0. We have

0 <= γf − e <= γf + γnf = γ(n+ 1)f, or equivalently

0 <= γ−1(n+ 1)−1(γf − e) <= f.

By the definition of r, this results in

γ−1(n+ 1)−1p(γf − e) = p(γ−1(n+ 1)−1(γf − e)) ≤ r(f), or equivalently

γ(n+ 1)r(f)− p(γf − e) ≥ 0.

We conclude that

g(γ · (nf, f) + (e,−e)) = γn(r(f) + ε) + p(e)

≥ γnr(f) + γ(r(f)− p(f)) + p(e)

= γ(n+ 1)r(f)− p(γf − e) ≥ 0,

proving that g is in fact positive. Our assumption f > 0 entails that (nf, f) is an order unitin A ⊕ A. By Proposition 2.4, g extends to a positive homomorphism g : A ⊕ A → R. Thefunction q : A→ R, defined by q(e) := g(e, 0), lies in (A∗)+. For every e ∈ A+, we have that

q(e) = g(e, 0) ≥ g(e, 0)− g(0, e) = g(e,−e) = p(e),

hence p ≤+ q. Then p+ ≤+ q by the positivity of q. As f lies in A+, we obtain

p+(f) ≤ q(f) = n−1g(nf, 0) ≤ n−1(g(nf, 0) + g(0, f))

= n−1g(nf, f) = n−1 · n(r(f) + ε) = r(f) + ε.

As ε was arbitrary, we get p+(f) ≤ r(f).Overall, we have shown that r = p+|A+ . In particular, r is additive.

36

3 Choquet simplices

3.8 Theorem Let X be a compact Hausdorff space, and let A ⊆ C(X;R) be a linear subspacecontaining 1. If (A∗,≤+) is a lattice, then (A,<=) is an interpolation group.

Proof. (cf. [Goo86, 11.3]) Endow A with the strict ordering. We will show that A satisfiesthe Riesz decomposition property which is equivalent to the Riesz interpolation property. Letf1, f2 ∈ A+. Consider the following sets:

Fi := {f ∈ A | 0 <= f <= fi}, for i = 1, 2 and

F := {f ∈ A | 0 <= f <= f1 + f2}.

We have to show that F ⊆ F1 +F2. This is trivial if either f1 or f2 equals zero. Therefore, wemay assume that 0 < f1, f2.Suppose that F1 + F2 is not a dense subset of F . Choose an element h ∈ F that does not liein the closure of F1 + F2. As F1 + F2 is convex, its closure is convex as well. According to theHahn-Banach theorem, there exists a continuous linear functional p ∈ A∗ satisfying

sup{p(f) | f ∈ F1 + F2} < p(h).

Define r : A+ → R by the formula

r(f) := sup{p(e) | e ∈ A, 0 <= e <= f}.

By Lemma 3.7, r is an additive function. However, the computation

r(f1) + r(f2) = sup(p(F1)) + sup(p(F2)) = sup(p(F1 + F2))

< p(h) ≤ sup(p(F )) = r(f1 + f2)

results in a contradiction, proving that F1 + F2 is a dense subset of F .To show that F ⊆ F1 + F2, let f ∈ F . We may assume that 0 6= f 6= f1 + f2, such that0 < f < f1 + f2. Choose ε > 0 such that ε < f1, f2 and ε < f < f1 + f2 − ε, then choose0 < δ < 1/2 such that 2δf < ε. We have that

2δε < ε < (1 + 2δ)f < f1 + f2 − ε+ 2δf < f1 + f2,

hence0 < (1 + 2δ)f − 2δε < f1 + f2,

in other words (1 + 2δ)f − 2δε ∈ F . As F1 +F2 is a dense subset of F , there exist g1 ∈ F1 andg2 ∈ F2 such that ‖(1 + 2δ)f − 2δε− g1 − g2‖ < 2δε. Let

h := ((1 + 2δ)f − 2δε− g1 − g2)/2,

hi := (1 + 2δ)−1(gi + δε+ h), for i = 1, 2.

37

3 Choquet simplices

As ‖h‖ < δε, we have that −δε < h < δε. For i = 1, 2, it follows that

0 < gi < gi + δε+ h < fi + 2δε < (1 + 2δ)fi.

Dividing by 1 + 2δ yields 0 < hi < fi, that is hi ∈ Fi. Then the computation

(1 + 2δ)(h1 + h2) = g1 + g2 + 2δε+ 2h = (1 + 2δ)f

shows that f = h1 + h2 ∈ F1 + F2.

3.9 Theorem For a compact convex set K, the following conditions are equivalent:(i) K is a Choquet simplex.(ii) (Aff(K), <=) is an interpolation group.(iii) (Aff(K),≤) is an interpolation group.

Proof. (cf. [Goo86, 11.4]) (i)⇒ (ii): As seen in Proposition 3.2, K is a base for the positive conein (Aff(K)∗,≤+). It follows from [Goo86, 7.3] that Aff(K)∗ is directed, or in other words that itis the linear space generated by its positive cone. So if K is assumed to be a Choquet simplex,then (Aff(K)∗,≤+) is a lattice. Theorem 3.8 states that (Aff(K), <=) is an interpolation group.(ii)⇒ (iii): Suppose that (Aff(K), <=) is an interpolation group, and let f1, f2, h1, h2 ∈ Aff(K)

such that f1, f2 ≤ h1, h2. We will inductively construct a sequence (gn)n∈N ⊆ Aff(K) satisfying

gn−1 − 1/2n <= gn <= gn−1 + 1/2n for all n > 1, as well as

f1 − 1/2n, f2 − 1/2n <= gn <= h1 + 1/2n, h2 + 1/2n for all n.

As f1 − 1, f2 − 1 <= h1 + 1, h2 + 1, we may choose g0 directly. Suppose that g0, . . . , gn−1 havealready been constructed. Note that

f1 − 1/2n, f2 − 1/2n, gn−1 − 1/2n <= h1 + 1/2n, h2 + 1/2n, gn−1 + 1/2n,

so we may again make use of interpolation to obtain gn with the desired properties. Thisfinishes the induction. By the first property, (gn)n∈N ⊆ Aff(K) is a Cauchy sequence. SinceAff(K) is a complete metric space, the sequence converges to some g ∈ Aff(K). The secondproperty guarantees that f1, f2 ≤ g ≤ h1, h2.(iii) ⇒ (i): If (Aff(K),≤) is an interpolation group, then K ∼= S(Aff(K), 1) is a Choquetsimplex according to Proposition 3.5.

Theorem 3.9 guarantees the existence of a large stock of continuous affine functions on aChoquet simplex K. However, a more efficient tool for constructing continuous affine functionson K is given by the following theorem, which was first proved by Edwards in [Edw65].

38

3 Choquet simplices

3.10 Theorem For a compact convex set K, the following conditions are equivalent:(i) K is a Choquet simplex.(ii) For any upper semicontinuous convex function f : K → [−∞,∞) and any lower semicon-

tinuous concave function h : K → (−∞,∞] satisfying f < h, there exists a continuousaffine function g ∈ Aff(K) such that f < g < h.

(iii) For any upper semicontinuous convex function f : K → [−∞,∞) and any lower semicon-tinuous concave function h : K → (−∞,∞] satisfying f ≤ h, there exists a continuousaffine function g ∈ Aff(K) such that f ≤ g ≤ h.

Proof. (cf. [Goo86, 11.12 and 11.13]) (i) ⇒ (ii): By Proposition 2.22, there exist a functionf ′ that is the pointwise maximum of finitely many continuous affine functions and a functionh′ that is the pointwise minimum of finitely many continuous affine functions such that f <f ′ < h′ < h. Theorem 3.9 implies that (Aff(K), <=) has interpolation. Thus, we can findg ∈ Aff(K) such that f ′ <= g <= h′, implying that f < g < h.(ii) ⇒ (iii): We will inductively construct a sequence (gn)n∈N ⊆ Aff(K) such that

gn−1 − 1/2n < gn < gn−1 + 1/2n for all n > 1, while also

f − 1/2n < gn < h+ 1/2n for all n.

As f − 1 < h+ 1, we can use (ii) to obtain g0 ∈ Aff(K) such that f − 1 < g0 < h+ 1. Supposethat we have constructed g0, . . . , gn−1 with the desired properties. Consider the functions

f ′ := max{f − 1/2n, gn−1 − 1/2n}h′ := min{h+ 1/2n, gn−1 + 1/2n}.

Notice that f ′ is upper semicountinuous and convex while h′ is lower semicontinuous andconcave such that f ′ < h′. Use (ii) to obtain gn ∈ Aff(K) such that f ′ < gn < h. Then gn isas desired, finishing the induction. The first property will guarantee that (gn)n∈N ⊆ Aff(K) isa Cauchy sequence. Since Aff(K) is a complete metric space, the sequence converges to someg ∈ Aff(K). The second property guarantees that f ≤ g ≤ h.(iii) ⇒ (i): Let f1, f2, h1, h2 ∈ Aff(K) such that f1, f2 ≤ h1, h2. Then the pointwise maximumof f1 and f2 is continuous and convex while the pointwise minimum of h1 and h2 is continuousand concave. We can therefore apply (iii) to obtain g ∈ Aff(K) such that f1, f2 ≤ g ≤ h1, h2.This shows that (Aff(K),≤) has interpolation. By Theorem 3.9, K is a Choquet simplex.

Theorem 3.10 is incredibly useful because lower semicontinuous concave functions (upper semi-continuous convex functions) appear naturally, for example as the pointwise minimum of lowersemicontinuous affine functions (pointwise minimum of upper semicontinuous affine functions).Our first application consists of showing that LAff(K) is a inf-semilattice whenever K is aChoquet simplex.

39

3 Choquet simplices

3.11 Lemma Let K be a Choquet simplex, and suppose that h : K → (−∞,∞] is a lowersemicontinuous concave function. There exists a biggest lower semicontinuous affine functionf ∈ LAff(K) such that f ≤ h.

Proof. Consider the set Λ := {e ∈ Aff(K) | e ≤ h}. The lower semicontinuity of h implies thath is bounded below, so Λ is nonempty. We claim that Λ is upward directed. Given e1, e2 ∈ Λ,their pointwise maximum is continuous and convex, satisfying max{e1, e2} ≤ h. By Theorem3.10, there exists e ∈ Aff(K) such that max{e1, e2} ≤ e ≤ h. It follows that e ∈ Λ ande1, e2 ≤ e.If f ∈ LAff(K) denotes the supremum of Λ, then f ≤ h. Let f ′ ∈ LAff(K) be another functionsuch that f ′ ≤ h. Choose an increasing net (ei)i∈I ⊆ Aff(K) with supremum f ′. For each i, theelement ei lies in Λ, hence ei ≤ f . Passing to the supremum shows that f ′ ≤ f , as desired.

3.12 Proposition For a compact convex set K, the following conditions are equivalent:(i) K is a Choquet simplex.(ii) LAff(K) has interpolation.(iii) LAff(K) is an inf-semilattice.

Proof. The implication (iii) ⇒ (ii) is trivial.(ii) ⇒ (i): Let f1, f2, h1, h2 ∈ Aff(K) such that f1, f2 < h1, h2. Choose an ε > 0 such thatf1 + ε, f2 + ε < h1, h2. Since LAff(K) has interpolation, we can choose h ∈ LAff(K) such thatf1 + ε, f2 + ε ≤ h ≤ h1, h2. But then f1, f2 � h, hence we can choose an e ∈ Aff(K) such thatf1, f2 < e < h ≤ h1, h2. We have shown that (Aff(K), <) has interpolation. It is fairly obviousthat this is equivalent to the condition that (Aff(K), <=) has interpolation. By Theorem 3.9,K is a Choquet simplex.(i) ⇒ (iii): Given h1, h2 ∈ LAff(K), their pointwise minimum is lower semicontinuous andconcave. By Lemma 3.11, there exists a biggest lower semicontinuous affine function f inLAff(K) such that f ≤ min{h1, h2}. Clearly, f is the infimum of h1 and h2 in LAff(K).

3.13 Lemma Let K be a Choquet simplex, and let F ⊆ ∂eK be a compact subset of theextreme boundary. Any continuous function g : F → R can be extended to a continuous affinefunction g ∈ Aff(K).

Proof. (cf. [Goo86, 11.14]) Define functions f : K → [−∞,∞), h : K → (−∞,∞] by setting

f(x) :=

{g(x) if x ∈ F−∞ if x /∈ F

and h(x) :=

{g(x) if x ∈ F∞ if x /∈ F

.

40

3 Choquet simplices

We claim that h is lower semicontinuous and concave. For any t ∈ R, the set

h−1((t,∞]) = {x ∈ K | h(x) > t}= {x ∈ K \ F | ∞ > t} ∪ {x ∈ F | g(x) > t}= (K \ F ) ∪ g−1((t,∞])

is open because F is compact and g is (lower semi-)continuous. Therefore h is lower semicon-tinuous. Let x, y ∈ K and t ∈ (0, 1). If tx+(1−t)y does not lie in F , then h(tx+(1−t)y) =∞,so the equation

h(tx+ (1− t)y) ≥ th(x) + (1− t)h(y)

is trivially satisfied. On the other hand, if tx + (1 − t)y lies in F ⊆ ∂eK, then x = y and thesame equation still holds trivially. This proves that h is lower semicontinuous and concave.Similarly, f is upper semicontinuous and convex and f ≤ h. We use Theorem 3.10 to obtaing ∈ Aff(K) such that f ≤ g ≤ h. As f |F = h|F = g, the function g has no choice but to be anextension of g.

3.3 Bauer simplices

Lemma 3.13 is particularly useful if ∂eK is compact to begin with. In this case, every continuousfunction on the extreme boundary extends to a continuous affine function on K. As a result,there is an abundance of continuous affine functions on K.

3.14 Definition A Bauer simplex is a Choquet simplex K such that ∂eK is compact.

3.15 Proposition Let K be a Bauer simplex. Every continuous function g : ∂eK → R extendsuniquely to a continuous affine function g : K → R. The restriction map Aff(K)→ C(∂eK;R)

is an isomorphism of partially ordered abelian groups.Similarly, every lower semicontinuous function h : ∂eK → (−∞,∞] extends uniquely to a lowersemicontinuous affine function h : K → (−∞,∞], and the restriction map LAff(K)→ Lsc(∂eK)

is an isomorphism of partially ordered abelian monoids.

Proof. That g extends to a continuous affine function g is an immediate consequence of Lemma3.13 when applied to F = ∂eK, and the uniqueness follows from Lemma 2.16. It is clear thatthe restriction map is a bijective OAG-morphism. Using Lemma 2.16 again, we deduce thatits inverse is order preserving.We can express h as the supremum of an increasing net (hi)i in C(∂eK;R). It is easy to checkthat h := supi hi is the unique extension to an element in LAff(K). The rest follows just likein the first case.

41

3 Choquet simplices

3.16 Proposition For a compact convex set K, the following conditions are equivalent:1) K is a Bauer simplex.2) Aff(K) is a lattice.3) LAff(K) is a lattice.

Proof. 1)⇒ 3): It was shown in Proposition 3.15 that the restriction map LAff(K)→ Lsc(∂eK)

is an order-isomorphism. Since Lsc(∂eK) is a lattice, it follows that LAff(K) is a lattice.3) ⇒ 2): It follows from Proposition 3.12 that K is a Choquet simplex. Given f, g ∈ Aff(K),their pointwise maximum h is continuous and convex. By definition, we have f, g ≤ sup{f, g}(where the supremum is taken in LAff(K)), or equivalently h ≤ sup{f, g}. Use Theorem 3.10to find e ∈ Aff(K) such that h ≤ e ≤ sup{f, g}. Since h ≤ e is again equivalent to f, g ≤ e,we obtain sup{f, g} ≤ e ≤ sup{f, g}. In particular, sup{f, g} = e is continuous. It followseasily from this that the supremum of f and g in Aff(K) exists and coincides with sup{f, g}.Therefore Aff(K) is a sup-semilattice. Then Aff(K) is also a lattice, where the infimum of fand g is given by inf{f, g} = − sup{−f,−g}.2) ⇒ 1): The assumption implies that Aff(K) has interpolation, or equivalently that K is aChoquet simplex (see 3.9). It is shown in [Goo86, 11.17] that a point x in a Choquet simplexK lies in ∂eK if and only if for all f, g ∈ Aff(K), there exists h ∈ Aff(K) such that h ≤ f, g

while also h(x) = min{f(x), g(x)}. In our case, this implies that

∂eK = {x ∈ K | inf{f, g}(x) = min{f(x), g(x)} for all f, g ∈ Aff(K)},

which is compact. Thus, K is a Bauer simplex.

The class of all Bauer simplices is well understood. We will see in the upcoming propositionthat up to isomorphism, Bauer simplices are exactly the compact convex sets of the formM+

1 (X), where X is a compact Hausdorff space.

3.17 Proposition If X is a compact Hausdorff space, then M+1 (X) is a Bauer simplex such

that ∂eM+1 (X) ∼= X. Conversely, if K is a Bauer simplex, then ∂eK is a compact Hausdorff

space such that K ∼= M+1 (∂eK).

Proof. Suppose that X is a compact Hausdorff space. By Corollary 3.6, M+1 (X) is a Choquet

simplex, and we have ∂eM+1 (X) ∼= X according to Proposition 2.2. In particular ∂eM+

1 (X) iscompact, hence M+

1 (X) is a Bauer simplex.Now suppose that K is a Bauer simplex. Then ∂eK is a compact Hausdorff space by definition.Proposition 3.15 implies that (Aff(K), 1) ∼= (C(∂eK;R), 1) as groups with order unit. We applyProposition 2.11 at the first step and Lemma 2.9 at the third step to deduce

K ∼= S(Aff(K), 1) ∼= S(C(∂eK;R), 1) ∼= M+1 (∂eK),

42

3 Choquet simplices

just as desired.

Remark So far, we have yet to encounter a single concrete example of a Choquet simplex Kthat is not a Bauer simplex. The so called Poulsen simplex is a metrizable Choquet simplexthat is far away from being a Bauer simplex: its exteme points form a dense subset. Also,every metrizable Choquet simplex is affinely homeomorphic to a closed face of K. For moreinformation about the Poulsen simplex, we refer to [LOS78].

3.18 Proposition IfK is a Bauer simplex, then LAff(K)+ is a domain. Let us denote the way-below relation in LAff(K)+ (LAff(K)) by �+ (�). For e ∈ LAff(K), we set e+ := sup{e, 0}.Given f, h ∈ LAff(K)+, we have f �+ h if and only if there exists g ∈ LAff(K) such thatf ≤ g+ while also g � h. If K is metrizable, then LAff(K)+ is countably based.

Proof. Since LAff(K) is a domain, the set Λ := {e ∈ LAff(K) | e � h} is upward directedwith supremum h. It follows that the set Λ+ := {e+ | e ∈ Λ} ⊆ LAff(K)+ is also upwarddirected with supremum h. So if f �+ h, then there exists an element g ∈ LAff(K) suchthat g � h and f ≤ g+. Conversely, assume that there exists a function g as above. LetD ⊆ LAff(K)+ be upward directed such that h ≤ supD. As g � h, there exists an elemente ∈ D such that g ≤ e. It follows that f ≤ g+ ≤ e, proving that f �+ h. We have shown thatthe claimed characterization of the way-below relation holds, and it follows from the first partthat LAff(K)+ is indeed a domain.If K is metrizable, then LAff(K) is countably based according to Proposition 2.24. If B isa countable basis for LAff(K), it is straightforward to check that B+ := {e+ | e ∈ B} is acountable basis for LAff(K)+. Thus, LAff(K)+ is countably based.

3.19 Proposition LetK be a metrizable Bauer simplex. Then LAff(K)+ is a countably basedCu-semigroup.

3.20 Lemma A compact Hausdorff space X is metrizable if and only ifM+1 (X) is metrizable.

Proof. The nontrivial implication is proved in [Goo86, 5.23].

If X is a metrizable compact Hausdorff space, it follows from Lemma 3.20 and Proposi-tion 3.19 that LAff(M+

1 (X))+ is a countably based Cu-semigroup. Since the natural mapρ : LAff(M+

1 (X))+ → Lsc(X)+ is a PoM-isomorphism, it follows that Lsc(X)+ is a countablybased Cu-semigroup and that ρ is a Cu-isomorphism. A similar statement holds for Lsc(X)0

++

(of course, both of these statements can be proved more directly, without making the detourto Bauer simplices). The situation is restated in the following corollary.

43

3 Choquet simplices

3.21 Corollary LetX be a metrizable compact Hausdorff space. Then Lsc(X)+ and Lsc(X)0++

are countably based Cu-semigroups. Moreover, the natural maps

LAff(M+1 (X))+ → Lsc(X)+ and

LAff(M+1 (X))0

++ → Lsc(X)0++

are Cu-isomorphisms.

44

4 A tensor product for compact convex setsand the main theorem

Recall that we have established a rough duality between compact convex sets and partiallyordered abelian groups with order unit. To each compact convex set K, we assign the groupwith order unit (Aff(K), 1), and to each group with order unit (G, u), we assign the compactconvex set S(G, u). There is a fairly obvious way to define a tensor product for groups withorder unit. This allows us to define a tensor product of two compact convex sets K1 and K2,by taking the tensor product of (Aff(K1), 1) and (Aff(K2), 1), then applying the state functor.

4.1 The tensor product of groups with order unit

We denote the category of abelian groups with group homomorphisms by AG. If G1, G2 and Hare abelian groups, a map G1×G2 → H is called AG-bimorphism if it is a group homomorphismin each variable. It is well known that the tensor product of abelian groups (Z-modules) exists.

4.1 Definition 1) Let G1, G2 and H be ordered abelian groups. A map ϕ : G1 × G2 → H

is called OAG-bimorphism if it is a AG-bimorphism and if ϕ(g1, g2) is positive whenever g1

and g2 are positive.2) Suppose that (G1, u1), (G2, u2) and (H, v) are partially ordered abelian groups with order

unit. A map ϕ : G1×G2 → H is called GOU-bimorphism if it is a OAG-bimorphism and ifϕ(u1, u2) = v. We may write ϕ : (G1, u1)× (G2, u2)→ (H, v) instead of ϕ : G1 ×G2 → H.

4.2 Proposition 1) The tensor product of partially ordered abelian groups exists, more pre-cisely: Let G1 and G2 be partially ordered abelian groups, and let (G1 ⊗AG G2, ω) be thetensor product of the underlying abelian groups. We can endow G1 ⊗AG G2 with a partialorder ≤ such that G1⊗OAGG2 := (G1⊗AGG2,≤) is a partially ordered abelian group, suchthat ω is a OAG-bimorphism and such that (G1 ⊗OAG G2, ω) is the tensor product of G1

and G2 as partially ordered abelian groups. Furthermore, the positive cone of G1 ⊗OAG G2

is generated by ω((G1)+, (G2)+).2) The tensor product of partially ordered abelian groups with order unit exists, more pre-

cisely: Let (G1, u1) and (G2, u2) be partially ordered abelian groups with order unit, and

45

4 A tensor product for compact convex sets and the main theorem

let (G1 ⊗OAG G2, ω) be their tensor product as partially ordered abelian groups with orderunit. Then (G1, u1)⊗GOU (G2, u2) := (G1⊗OAGG2, ω(u1, u2)) is a partially ordered abeliangroup with order unit, and ((G1, u1)⊗GOU (G2, u2), ω) is the tensor product of (G1, u1) and(G2, u2) as partially ordered abelian groups with order unit.

Proof. For 1), let C ⊆ G1 ⊗AG G2 be the cone generated by ω((G1)+, (G2)+). The difficultylies in showing that C is strict. However, this is done in [GH86, 2.1]. It follows that the orderinduced by C is a partial order. The rest of 1) is straightforward and will be omitted. It isclear that 2) holds.

4.3 Proposition Let (G1, u1) and (G2, u2) be two groups with order unit. If s1 and s2 arestates on (G1, u1) and (G2, u2) respectively, there is a unique state on (G1, u1)⊗GOU (G2, u2)

which maps an elementary tensor x� y to s1(x) · s2(y). The map

S(G1, u1)× S(G2, u2)→ S((G1, u1)⊗GOU (G2, u2))

is jointly continuous and affine in each variable. The continuous and affine maps

S((G1, u1)⊗GOU (G2, u2))→ S(G1, u1), s 7→ s(_� u2)

S((G1, u1)⊗GOU (G2, u2))→ S(G2, u2), s 7→ s(u1 �_)

are referred to as the projections onto the first and the second variable respectively. Thecomposition

S(G1, u1)× S(G2, u2)→ S((G1, u1)⊗GOU (G2, u2))→ S(G1, u1)× S(G2, u2)

equals the identity. In particular, the first map is injective.

Proof. Let s1 and s2 be states on (G1, u1) and (G2, u2) respectively. The map

(G1, u1)× (G2, u2)→ (R, 1), (x, y) 7→ s1(x) · s2(y)

is a GOU-bimorphism. The corresponding GOU-morphism (G1, u1)⊗GOU (G2, u2)→ (R, 1) isthe desired state. Let us denote the map

S(G1, u1)× S(G2, u2)→ S((G1, u1)⊗GOU (G2, u2))

by ϕ. Fix some state s2 ∈ S(G2, u2). Let s1, s′1 ∈ S(G1, u1), t ∈ [0, 1]. For every x ∈ G1 and

y ∈ G2, we have that

ϕ(t · s1 + (1− t) · s′1, s2)(x� y) = (t · s1 + (1− t) · s′1)(x) · s2(y)

= t · s1(x) · s2(y) + (1− t) · s′1(x) · s2(y)

= (t · ϕ(s1, s2) + (1− t) · ϕ(s′1, s2))(x� y),

46


hence ϕ(t · s1 + (1 − t) · s′1, s2) = t · ϕ(s1, s2) + (1 − t) · ϕ(s′1, s2) by uniqueness. Since s1, s′1and t were arbitrary, the map ϕ(_, s2) is affine. Similarly, ϕ is affine in the second variable.For n ∈ {1, 2}, let (sn,λ)λ∈Λ ⊆ S(Gn, un) be a net converging to some sn ∈ S(G1, u1). By thecompactness of S((Gn, un)⊗GOU(G2, u2)), the net (ϕ(s1,λ, s2,λ))λ∈Λ has at least one convergentsubnet. Now suppose that (ϕ(s1,α(i), s2,α(i)))i∈I is an arbitrary convergent subnet, and letf ∈ S((G1, u1)⊗GOU (G2, u2)) denote the limit. For every x ∈ G1 and y ∈ G2, we have that

f(x� y) = (limiϕ(s1,α(i), s2,α(i)))(x� y) = lim

i(ϕ(s1,α(i), s2,α(i))(x� y))

= limi

(s1,α(i)(x) · s2,α(i)(y)) = s1(x) · s2(y) = ϕ(s1, s2)(x� y).

Using uniqueness again, we deduce f = ϕ(s1, s2). We have shown that the net (ϕ(s1,λ, s2,λ))λ∈Λ

has exactly one limit point, namely ϕ(s1, s2). Since S((G1, u1)⊗GOU (G2, u2)) is compact, thenet must converge to that point. We have shown that ϕ is jointly continuous.It is easy to check that the projections

S((G1, u1)⊗GOU (G2, u2))→ S(G1, u1), s 7→ s(_� u2)

S((G1, u1)⊗GOU (G2, u2))→ S(G2, u2), s 7→ s(u1 �_)

are well defined, continuous and affine. It remains to show that the composition

S(G1, u1)× S(G2, u2)→ S((G1, u1)⊗GOU (G2, u2))→ S(G1, u1)× S(G2, u2)

equals the identity. For s1 ∈ S(G1, u1) and s2 ∈ S(G2, u2), we use s to denote their image inS((G1, u1) ⊗GOU (G2, u2)). Then s is the unique state that maps an elementary tensor x � yto the value s1(x) · s2(y). For all x ∈ G1, we have that

s(x� u2) = s1(x) · s2(u2) = s1(x) · 1 = s1(x),

hence s(_ � u2) = s1. Similarly s(u1 � _) = s2, so the second map sends s back to (s1, s2),finishing the proof.

4.2 Dimension groups

Before we move on to the tensor product of compact convex sets, let us briefly turn our attentionto a structural result by Effros, Handelman and Shen. Apart from 0, the ‘simplest’ orderedabelian group one can think of is Z or, more general, finite powers of Z. The category OAGhas inductive limits which are based on the algebraic inductive limit. The next step would beto consider the class of inductive limits of finite powers of Z. Theorem 4.6 states that this classcoincides with the class of all dimension groups. There is one crucial point in the proof of themain theorem where said theorem is employed.

47


A partially ordered abelian group G is said to be unperforated if for all g ∈ G, the assumptionthat some nonzero multiple of g is positive implies that g is positive.

4.4 Definition A partially ordered abelian group G is called simplicial if there exists n ∈ Nsuch that G is isomorphic to Zn with the usual order and addition.We say thatG is a dimension group if it is directed, unperforated and has the Riesz interpolationproperty.

4.5 Example If K is a Choquet simplex, then both (Aff(K),≤) and (Aff(K), <=) are dimen-sion groups.

4.6 Theorem ([EHS80, 2.2]) A partially ordered abelian group is a dimension group if andonly if it is the inductive limit of simplicial groups.

4.3 A tensor product for compact convex sets

4.7 Lemma Let K1 and K2 be compact convex sets. For any map ϕ : Aff(K1)×Aff(K2)→ R,the following are equivalent:1) ϕ is a OAG-bimorphism (Aff(K1),≤)× (Aff(K2),≤)→ (R,≤).2) ϕ is a OAG-bimorphism (Aff(K1), <=)× (Aff(K2), <=)→ (R,≤).In particular, we see that

S((Aff(K1),≤, 1)⊗GOU (Aff(K2),≤, 1)) = S((Aff(K1), <=, 1)⊗GOU (Aff(K2), <=, 1)).

Proof. The implication 1) ⇒ 2) is trivial, so let us show that the implication 2) ⇒ 1) holds.Let f ∈ Aff(K1)+, g ∈ Aff(K2)+. For every n ∈ N∗, we have that

ϕ(f, g) = ϕ(f + 1n , g + 1

n)− ϕ(f + 1n ,

1n)− ϕ( 1

n , g + 1n) + ϕ( 1

n ,1n)

= ϕ(f + 1n , g + 1

n)− 1nϕ(f + 1

n , 1)− 1nϕ(1, g + 1

n) + ϕ( 1n ,

1n)

≥ 0− 1nϕ(f + 2, 1)− 1

nϕ(1, g + 2) + 0 = − 1n(ϕ(f + 2, 1) + ϕ(1, g + 2)).

Passing to the supremum yields ϕ(f, g) ≥ 0, as desired.

4.8 Definition For two compact convex sets K1 and K2, we define their tensor product as

K1 ⊗K2 := S((Aff(K1),≤, 1)⊗GOU (Aff(K2),≤, 1))

= S((Aff(K1), <=, 1)⊗GOU (Aff(K2), <=, 1)).

The equality is given by Lemma 4.7.

48


Let K1 and K2 be two compact convex sets. If we apply Proposition 4.3 to (Aff(K1), 1) and(Aff(K2), 1), we obtain a map

K1 ×K2∼=−→ S(Aff(K1), 1)× S(Aff(K2), 1)→ S((Aff(K1), 1)⊗GOU (Aff(K2), 1)) = K1 ⊗K2

that is jointly continuous, affine in each variable and injective. It maps (x, y) ∈ K1 × K2 tothe unique state x⊗ y on (Aff(K1), 1)⊗GOU (Aff(K2), 1) that maps an elementary tensor f � gto f(x) · f(y).

Remark The tensor product defined in 4.8 is known as the biprojective tensor product of K1

and K2. There are two more notions of tensor products of compact convex sets, the so calledbiinjective tensor product and the so called projective tensor product. In general, these tensorproducts disagree. However, they coincide if either K1 or K2 is a Choquet simplex. Theprojective tensor product of K1 and K2 has the universal property that continuous and affinemaps from the projective tensor product into a different compact convex set K correspond tomaps K1 × K2 → K that are continuous and affine in each variable. For more informationabout these tensor products, we refer to [NP69] (in that paper, the biprojective tensor productis denoted by K1�K2).

4.9 Theorem Let K1 and K2 be compact convex sets. If one of them is a Choquet simplex,then

∂e(K1 ⊗K2) = {x⊗ y | x ∈ ∂eK1, y ∈ ∂eK2}.

In particular ∂e(K1 ⊗K2) is homeomorphic to ∂eK1 × ∂eK2.

Proof. The first statement is proved in [NP69, 2.1]. It follows from the first statement that themap ∂eK1 × ∂eK2 → ∂e(K1 ⊗K2), which maps (x, y) to x⊗ y, is well defined and surjective.We know already that tensoring is continuous and injective. Since we are only dealing withcompact Hausdorff spaces, this map is a homeomorphism.

Let us prove that the tensor product of two Choquet simplices is again a Choquet simplex. Onthe level of groups with order unit, one would hope that the tensor product of two interpolationgroups is an interpolation group. Unfortunately, this turns out to be wrong. In [Weh96, 1.5],Wehrung gives an example of two (torsion-free, directed) interpolation groups G and H suchthat G ⊗OAG H does not have interpolation. However, if we assume that one of the orderedgroups is unperforated, then the tensor product has interpolation, as shown in Proposition4.11, which is taken from [Weh96, 1.7].

4.10 Proposition ([Weh96, 1.2]) Let G and H be partially ordered abelian groups, andconsider the directed subgroups Gdir := G+ − G+ and Hdir := H+ −H+. If either G or H isunperforated, then (G⊗OAG H)+

∼= (Gdir ⊗OAG Hdir)+.

49


4.11 Proposition Suppose that G and H are two interpolation groups, one of which is un-perforated. Then G⊗OAG H has interpolation.

Proof. Assume that G is unperforated. Recall that G⊗OAG H has interpolation if and only if(G⊗OAG H)+ has the Riesz decomposition property. Using Proposition 4.10, we may assumethat both G and H are directed. In this case, G is a dimension group. By Theorem 4.6, wemay express G as an inductive limit of simplicial groups Gi. It is stated in [GH86, 2.2] that thefunctor _ ⊗OAG H preserves inductive limits. It follows that G ⊗OAG H ∼= lim−→(Gi ⊗OAG H).For each i, the ordered group Gi ⊗OAG H has interpolation (since Zn ⊗OAG H ∼= Hn and Hhas interpolation). It is not hard to see that G⊗OAG H must have interpolation as well.

4.12 Corollary If K1 and K2 are Choquet simplices, then K1 ⊗K2 is a Choquet simplex.

Proof. Since both Aff(K1) and Aff(K2) are unperforated interpolation groups, it follows fromProposition 4.11 that Aff(K1)⊗OAGAff(K2) is an interpolation group. Consequently,K1⊗K2 =

S((Aff(K1), 1)⊗GOU (Aff(K2), 1)) is a Choquet simplex.

4.13 Corollary If K1 and K2 are Bauer simplices, then K1 ⊗K2 is a Bauer simplex. In factK1 ⊗K2

∼= M+1 (∂eK1 × ∂eK2). (We may write this as M+

1 (X)⊗M+1 (Y ) ∼= M+

1 (X × Y ).)

Proof. We know from Corollary 4.12 that K1⊗K2 is a Choquet simplex. Since K1 and K2 areBauer simplices, the product ∂eK1 × ∂eK2, which according to Theorem 4.9 is homeomorphicto ∂e(K1 ⊗K2), is compact. Therefore K1 ⊗K2 is a Bauer simplex. We use Proposition 3.17at the first step to deduce K1 ⊗K2

∼= M+1 (∂e(K1 ⊗K2)) ∼= M+

1 (∂eK1 × ∂eK2).

4.4 The main theorem

Let G and H be partially ordered abelian groups. The map G+ × H+ → (G ⊗OAG H)+,(g, h) 7→ g � h is a PoM-bimorphism that induces a natural PoM-morphism

G+ ⊗PoM H+ → (G⊗OAG H)+,

which by the definition of the positive cone in G⊗OAG H is always surjective. In general, thismap is not a PoM-isomorphism (or equivalently injective). The problem is that G+ ⊗PoM H+

may fail to be cancellative. A counterexample can be found in [APT18], just above sectionB.4.For compact convex sets K1 and K2, consider the following special cases:

Aff(K1)+ ⊗PoM Aff(K2)+ → ((Aff(K1),≤)⊗OAG (Aff(K2),≤))+,

Aff(K1)0++ ⊗PoM Aff(K2)0

++ → ((Aff(K1), <=)⊗OAG (Aff(K2), <=))+.

50


Also, by definition of K1 ⊗ K2, we have the following two GOU-morphisms (the underlyingmaps are the same):

ι : (Aff(K1),≤, 1)⊗GOU (Aff(K2),≤, 1)→ (Aff(K1 ⊗K2),≤, 1),

ι : (Aff(K1), <=, 1)⊗GOU (Aff(K2), <=, 1)→ (Aff(K1 ⊗K2),≤, 1).

For f ∈ Aff(K1) and g ∈ Aff(K2), we set f⊗g := ι(f�g). Notice that (f⊗g)(x⊗y) = f(x)·g(y)

holds for all x ∈ K1 and y ∈ K2.In order to prove the main theorem, we study the maps

π : Aff(K1)0++ ⊗PoM Aff(K2)0

++ → ((Aff(K1), <=)⊗OAG (Aff(K2), <=))+,


In the case of arbitrary groups with order unit, the natural map ι is rarely surjective. We willshow that that the image of ι is a dense subspace of Aff(K1⊗K2), which is good enough. Onewould hope that ι, at least as a map

ι : (Aff(K1),≤, 1)⊗GOU (Aff(K2),≤, 1)→ (Aff(K1 ⊗K2),≤, 1),

is an embedding. This is in general not the case, not even for nice compact convex sets. Forexample, by setting K1 = K2 = M+

1 ([0, 1]), ι may be identified with the map

C([0, 1];R)⊗OAG C([0, 1];R)→ C([0, 1]2;R),

which is not an embedding (see [Fre72, 4.7]). Fortunately, we can prove that ι is not far awayfrom being an embedding. Finally, we would like π to be an isomorphism. While it is notknown to me whether or not this is the case for arbitrary compact convex sets, we will provethat this is true under the assumption that K1 or K2 is a Choquet simplex.For the remainder of this chapter, we denote the image of ι by A. Also, we denote the imageof ι ◦ π by B.

4.14 Lemma Let K1 and K2 be compact convex sets, and consider the map

ω : Aff(K1)×Aff(K2)→ Aff(K1 ⊗K2), (f, g) 7→ f ⊗ g.

For all f, f ′ ∈ Aff(K1), g, g′ ∈ Aff(K2) and λ, µ ∈ R, the following holds:1) 1⊗ 1 = 1,2) (f + f ′)⊗ g = f ⊗ g + f ′ ⊗ g, and f ⊗ (g + g′) = f ⊗ g + f ⊗ g′,3) f ⊗ g ≥ 0 if f, g ≥ 0, and f ⊗ g > 0 if f, g > 0,4) (λf)⊗ (µg) = λµ(f ⊗ g),

51


5) ω is continuous in each variable.Furthermore, if either K1 or K2 is a Choquet simplex, it is true that:6) ‖f ⊗ g‖ ≤ ‖f‖ · ‖g‖,7) ω is jointly continuous.

Proof. Points 1), 2) and 3) follow directly from the definition of GOU-bimorphism. Point 4) isclear for λ, µ ∈ Q. It will follow from 5) that the formula is true for arbitrary λ, µ ∈ R. In orderto prove 5), let h ∈ Aff(K2). For now, let us assume that h is positive. Let e ∈ Aff(K1), ε > 0.Choose δ ∈ Q, δ > 0 such that δ · ‖1⊗ h‖ ≤ ε. If e′ lies in Aff(K1) and satisfies ‖e− e′‖ ≤ δ,we have that −δ ≤ e− e′ ≤ δ. Since h is positive, we obtain

(e− e′)⊗ h ≤ δ ⊗ h = δ(1⊗ h) ≤ δ · ‖1⊗ h‖ ≤ ε,

and similarly −ε ≤ (e − e′) ⊗ h, whence ‖e ⊗ h − e′ ⊗ h‖ ≤ ε. Note that we used 4) at thesecond step. This is not circular reaoning however, since δ is rational. We have shown that themap ω(_, h) is continuous if h is positive. The general case follows by writing h as a differenceof positive elements. Thus, ω is continuous in the first variable. By the symmetry of thisargument, ω is continuous in each variable.Now suppose that either K1 or K2 is a Choquet simplex. In this case, we have that

∂e(K1 ⊗K2) = {x⊗ y | x ∈ ∂eK1, y ∈ ∂eK2},

according to Theorem 4.9. It follows that

‖f ⊗ g‖ = maxz∈K1⊗K2

|(f ⊗ g)(z)| = maxz∈∂e(K1⊗K2)

|(f ⊗ g)(z)|

= maxx∈∂eK1,y∈∂eK2

|(f ⊗ g)(x⊗ y)| = maxx∈∂eK1,y∈∂eK2

|f(x) · g(y)|

≤ ( maxx∈∂eK1

|f(x)|) · ( maxy∈∂eK2

|g(y)|) = (maxx∈K1

|f(x)|) · (maxy∈K2

|g(y)|) = ‖f‖ · ‖g‖,

which proves 6). Then 7) will easily follow from this.

The following theorem is due to Jellett (see [Jel68, p. 224]). An alternative proof can be foundin [Goo86, 7.4].

4.15 Theorem Let K be a compact convex set, and let V ⊆ Aff(K) be a linear subspace. IfV contains 1 and separates the points of K, then V is dense in Aff(K).

It follows from Lemma 4.14 that A is a subspace of Aff(K1 ⊗ K2) containing 1. It is trivialthat A separates the points of K1 ⊗K2. Thus, we obtain the following corollary.

4.16 Corollary A is a dense subspace of Aff(K1 ⊗K2).

52


4.17 Corollary If K1 and K2 are two metrizable compact convex sets, then K1 ⊗ K2 ismetrizable.

Proof. Since K1 and K2 are metrizable, Aff(K1) and Aff(K2) are separable according to Propo-sition 2.24. Suppose that B1 ⊆ Aff(K1) and B2 ⊆ Aff(K2) are countable dense subsets. Weclaim that the countable set

B :=

{n∑i=0

f(1)i ⊗ f

(2)i | n ∈ N and f (j)

i ∈ Bj for all i and j

}

is dense in Aff(K1⊗K2). The map Aff(K1)×Aff(K2)→ Aff(K1⊗K2), (f, g) 7→ f⊗g is contin-uous in each variable, according to Lemma 4.14. It follows from this that we can approximateeach elementary tensor in Aff(K1 ⊗ K2) by elements in B. Thus, we can approximate eachelement in A by elements in B. Since A lies dense in Aff(K1 ⊗K2), B must also be a densesubset of Aff(K1 ⊗ K2). We have shown that Aff(K1 ⊗ K2) is separable. Using Proposition2.24 a second time, we deduce that K1 ⊗K2 is metrizable.

4.18 Proposition Let (G, u) be a nonzero, unperforated partially ordered abelian group withorder unit, and consider the natural map ι : (G, u) → (Aff(S(G, u)), 1). For g ∈ G, we haveι(g) > 0 if and only if g is an order unit in G.

Proof. [Goo86, 4.13].

4.19 Lemma Let G be a partially ordered vector space, and let H be a partially orderedabelian group. There exists a unique OAG-bimorphism

m : R× (G⊗OAG H)→ G⊗OAG H,

that satisfies m(t, g � h) = (t · g) � h for all t ∈ R, g ∈ G and h ∈ H. When equipped withthis scalar multiplication, G⊗OAG H carries the structure of a partially ordered vector space.In particular, G⊗OAG H is unperforated.

Proof. Let t ∈ R. The map

G×H → G⊗OAG H, (g, h) 7→ (t · g)� h

is a AG-bimorphism. If t is positive, then this map is a OAG-bimorphism. Let us denote thecorresponding AG-morphism G ⊗OAG H → G ⊗OAG H by mt. Again, if t is positive, thenmt is a OAG-morphism. Then define m : R × (G ⊗OAG H) → G ⊗OAG H by the formulam(t, x) := mt(x). Using the universal property of the tensor product, one checks that theequations m1 = id, mt+t′ = mt + mt′ and mt·t′ = mt ◦ mt′ are satisfied for all t, t′ ∈ R. It

53


follows from this that m has all the properties of a scalar multiplication. Moreover, m is anOAG-bimorphism since mt is a OAG-morphism whenever t is positive. The claimed uniquenessof m is clear. Thus, G ⊗OAG H can be considered as a partially ordered vector space. It istrivial that any partially ordered vector space is unperforated.

Now that we understand ι well enough, let us turn our attention to the map


++ → ((Aff(K1), <=)⊗OAG (Aff(K2), <=))+.

More general, we try to find a condition for which the natural mapG+⊗PoMH+ → (G⊗OAGH)+

is a PoM-isomorphism.For the upcoming Lemma, we need the following: given an abelian monoid M , let Gr(M)

denote the Grothendieck completion. The universal map δ : M → Gr(M) is an embedding ifand only if M is cancellative (i.e. x + z = y + z implies that x = y, for all x, y, z ∈ M). IfM is both cancellative and conical, it follows that δ(M) is a conical submonoid of Gr(M). Inthis case, we can consider the partially ordered abelian group OGr(M) := (Gr(M),≤δ(M)).Observe that OGr(M) is directed. The map δ : M → OGr(M)+ is a AM-isomorphism. If Mis a PoM, then δ is a PoM-isomorphism if and only if M is algebraically ordered. Both thecategory AM and the category PoM have inductive limits which are based on the algebraicinductive limit. Let OAGdir denote the full subcategory of OAG, consisting of directed orderedabelian groups. It is not too hard to see that OAGdir is closed under the inductive limit inOAG.

4.20 Lemma Let H be a partially ordered abelian group, and let N be a positively orderedmonoid.1) The functor _⊗OAG H : OAG→ OAG preserves inductive limits.2) The functor _⊗PoM N : PoM→ PoM preserves inductive limits.3) The functor (_)+ : OAGdir → PoM preserves inductive limits.

Proof. 1): This is stated in [GH86, 2.2].2): The functor _ ⊗PoM N is left adjoint to the functor HomPoM(N,_) : PoM → PoM. Itfollows from general category theory that _ ⊗PoM N is cocontinuous. Since inductive limitsare just a special case of a colimit, _⊗PoM N must preserve inductive limits.3): Let (Gi)i be an inductive system in OAGdir, and let M := lim−→(Gi)+ be the inductive limitof the positive cones in PoM. Suppose that x, y, z ∈ M satisfy x + z = y + z. It follows fromthe construction of the inductive PoM-limit that there exists an index k and representativesx′, y′, z′ ∈ (Gk)+ for x, y, z satisfying x′ + z′ = y′ + z′. Since (Gk)+ is cancellative, we obtainthat x′ = y′, whence x = y. Thus, M is cancellative. Similarly, one can show that M is conical

54


and algebraically ordered. It follows that M and OGr(M)+ are naturally isomorphic as PoMs.The following bijections are natural in partially ordered abelian groups A:

HomOAG(OGr(lim−→(Gi)+), A) ∼= HomAM(lim−→(Gi)+, A+) ∼= lim←−HomAM((Gi)+, A+)

∼= lim←−HomOAG(Gi, A) ∼= HomOAG(lim−→Gi, A).

It follows that OGr(lim−→(Gi)+) ∼= lim−→Gi. Note that the third step relied on the fact that eachGi is directed. If we combine all the isomorphisms, we obtain

lim−→(Gi)+∼= OGr(lim−→(Gi)+)+

∼= (lim−→Gi)+

as PoMs, which is the desired result.

4.21 Proposition Let G and H be partially ordered abelian groups, and suppose that one ofthem is unperforated and has interpolation. Then the natural map

G+ ⊗PoM H+ → (G⊗OAG H)+

is a PoM-isomorphism.

Proof. Since the tensor product is commutative, we may assume without loss of generality thatG is unperforated and has interpolation. Then, using Proposition 4.10, we may assume withoutloss of generality that both G and H are directed. In this case, G is a dimension group. WriteG as the inductive limit of a system (Gi)i of simplicial groups. For any n ∈ N, we have thatZn⊗OAGH ∼= Hn. As H is directed, it follows that Gi⊗OAGH is directed for each i. ApplyingLemma 4.20, we find that• lim−→(Gi ⊗OAG H) ∼= G⊗OAG H,• lim−→(Gi ⊗OAG H)+

∼= (G⊗OAG H)+,• lim−→(Gi)+

∼= G+,• lim−→((Gi)+ ⊗PoM H+) ∼= G+ ⊗PoM H+.

For every i, the map (Gi)+ ⊗PoM H+ → (Gi ⊗OAG H)+ is an isomorphism, because Gi is asimplicial group. Using that this isomorphism is natural at the second step, we deduce

G+ ⊗PoM H+∼= lim−→((Gi)+ ⊗PoM H+) ∼= lim−→(Gi ⊗OAG H)+

∼= (G⊗OAG H)+.

The composition of these PoM-isomorphisms is given by the natural map.

Let us summarize our findings in the following corollary.

55


4.22 Corollary Let K1 and K2 be compact convex sets, and consider


++ → ((Aff(K1), <=)⊗OAG (Aff(K2), <=))+,


The image of ι is a dense subspace of Aff(K1⊗K2). SetG := (Aff(K1), <=)⊗OAG(Aff(K2), <=).If x, y ∈ G satisfy ι(x) < ι(y), then y − x lies in G+. If K1 or K2 is a Choquet simplex, thenπ is a PoM-isomorphism.Analogous statements hold for

Aff(K1)+ ⊗PoM Aff(K2)+ → ((Aff(K1),≤)⊗OAG (Aff(K2),≤))+,

(Aff(K1),≤, 1)⊗GOU (Aff(K2),≤, 1)→ (Aff(K1 ⊗K2),≤, 1).

4.23 Lemma Let S be a Cu-semigroup and suppose that D ⊆ S is a dense submonoid, that isfor every s1, s2 ∈ S with s1 � s2, there exists d ∈ D such that s1 ≤ d� s2. The restriction ofthe way below relation in S toD is an auxiliary relation, and (D,�) is a W-semigroup. If T is aCu-semigroup, then every W-morphism (D,�)→ (T,�) extends uniquely to a Cu-morphismS → T . In particular γ(D,�) ∼= S.

Proof. It is clear that that the restriction of� onto D is an auxiliary relation, and that (D,�)

satisfies (W3). The axioms (W1), (W2) and (W4) will be satisfied since D is dense in S.Suppose that T is a Cu-semigroup and that ϕ : (D,�) → (T,�) is a W-morphism. Givens ∈ S, choose an increasing sequence (dn)n∈N ⊆ D with supremum s. Then (ϕ(dn))n∈N isan increasing sequence in T , so it has a supremum which we denote by ϕ(s). We will showthat this value does not depend on the choice of the increasing sequence. Let (en)n∈N ⊆ D beanother increasing sequence with supremum s. For k ∈ N, let t ∈ T such that t � ϕ(dk). Bythe continuity of ϕ, we may choose an element a ∈ D such that a � dk while also t ≤ ϕ(a).As dk ≤ supn dn = s = supn en, there exists an index i such that a ≤ ei. It follows thatt ≤ ϕ(a) ≤ ϕ(ei) ≤ supn ϕ(en). Since t was arbitrary, we can deduce that ϕ(dk) ≤ supn ϕ(en).Passing to the supremum, we get supn ϕ(dn) ≤ supn ϕ(en). This argument is symmetric, hencesupn ϕ(dn) = supn ϕ(en).By the argument above, the map ϕ : S → T is well defined. It is clear that ϕ is an extensionof ϕ. A straightforward, but tedious, computation shows that ϕ is a Cu-morphism. We omitthis. Uniqueness follows from the fact that D is dense in S.

Remark In the situation of Lemma 4.23, the restriction of the way-below relation in S toD agrees with the way-below relation in D, because D is dense in S. One application ofLemma 4.23 is the following: If K is a metrizable compact convex set, then (Aff(K)0

++,�) isa W-semigroup satisfying γ(Aff(K)0

++,�) ∼= LAff(K)0++.

56


We are now in the position to prove the main theorem. Most of the work is done in the followingLemma. Recall that we use A to denote the image of ι and B to denote the image of ι ◦ π.

4.24 Lemma Let K1 and K2 be compact convex sets. The following holds:1) B = A ∩Aff(K1 ⊗K2)0

++,2) B is a dense submonoid of LAff(K1 ⊗K2)0

++.Now suppose that both K1 and K2 are metrizable, and that either K1 or K2 is a Choquetsimplex. Then:3) (B,�) is a W-semigroup, and we have that

γ((Aff(K1)0++,�)⊗PreW (Aff(K2)0

++,�)) ∼= γ(B,�) ∼= LAff(K1 ⊗K2)0++.

Proof. Set G := (Aff(K1), <=)⊗OAG (Aff(K2), <=), and P := Aff(K1)0++ ⊗PoM Aff(K2)0

++.1) It follows from Lemma 4.14 that B is contained in A ∩ Aff(K1 ⊗K2)0

++. For the converseinclusion, let f ∈ A ∩ Aff(K1 ⊗ K2)0

++, and assume that f > 0. Choose f ′ ∈ G such thatι(f ′) = f . Then f ′ lies in G+, so the surjectivity of π allows us to find an element f ′′ ∈ P suchthat π(f ′′) = f ′. But then (ι ◦ π)(f ′′) = f , so f lies in B.2) Clearly B is a submonoid of LAff(K1 ⊗ K2)0

++. Suppose that f, h ∈ LAff(K1 ⊗ K2)0++

satisfy f � h, and assume that h 6= 0. Choose e ∈ Aff(K1 ⊗ K2) and ε > 0 such thatf < e− ε < e < e+ ε < h. Since A is dense in Aff(K1⊗K2), we can choose an element e′ ∈ Asuch that ‖e′− e‖ < ε. This entails that −ε < e′− e < ε, implying that f < e′ < h. Now e′ liesin A ∩Aff(K1 ⊗K2)0

++, which is equal to B by part 1). Additionally, f ≤ e′ � h, as desired.3) Since both K1 and K2 are assumed to be metrizable, K1 ⊗K2 is metrizable according toCorollary 4.17. Then LAff(K1 ⊗K2)0

++ is a Cu-semigroup, and it follows from 2) and Lemma4.23 that (B,�) is a W-semigroup satisfying γ(B,�) ∼= LAff(K1 ⊗K2)0

++. If we assume thateither K1 or K2 is a Choquet simplex, then π : P → G+ is a PoM-isomorphism. Using Lemma4.14, one can check that the map

ψ : (Aff(K1)0++,�)× (Aff(K2)0

++,�)→ (B,�), ψ(f, g) := f ⊗ g

is a W-bimorphism. By the universal property of the tensor product, there exists a uniqueW-morphism ψ : (Aff(K1)0

++,�)⊗PreW (Aff(K2)0++,�)→ (B,�) such that the diagram

(Aff(K1)0++,�)× (Aff(K2)0

++,�) (Aff(K1)0++,�)⊗PreW (Aff(K2)0

++,�)

(B,�)ψ

ψ

commutes. However, the same diagram commutes if we replace ψ by ι ◦π. Using the universalproperty of the tensor product on the PoM-level, we conclude that ψ = ι ◦ π. Thus, ι ◦ π is aW-morphism.

57


Set T := (Aff(K1)0++,�) ⊗PreW (Aff(K2)0

++,�). Let S be a Cu-semigroup, and supposethat ϕ : T → (S,�) is a W-morphism. We claim that there exists a unique W-morphismϕ : (B,�) → (S,�) such that ϕ = ϕ ◦ ι ◦ π. Define ϕ as follows: Given c ∈ B, choose c′ ∈ Psuch that (ι ◦ π)(c′) = c. Then set ϕ(c) := ϕ(c′). Let us show that this value does not dependon the choice of c′. In general, Lemma 4.14 implies that f ⊗ g = 0, for f ∈ Aff(K1)0

++ andg ∈ Aff(K2)0

++, can only occur if f = 0 or g = 0. It follows that the only choice for c′ in the casec = 0 is c′ = 0. Now assume that c > 0. Let c′′ ∈ P be another element such that (ι◦π)(c′′) = c.Let s ∈ S such that s� ϕ(c′). By the continuity condition of ϕ, there exists an element e′ ∈ Tsuch that e′ ≺ c′ (where ≺ denotes the auxiliary relation in T ) while also s ≤ ϕ(e′). As ι ◦ πpreserves the auxiliary relation, we see that (ι ◦ π)(e′) � (ι ◦ π)(c′) = (ι ◦ π)(c′′) 6= 0, hence(ι ◦ β)(e′) < (ι ◦ β)(c′′). We use Corollary 4.22 to deduce β(e′) ≤ β(c′′), and then e′ ≤ c′′.It follows that s ≤ ϕ(e′) ≤ ϕ(c′′). Passing to the supremum yields ϕ(c′) ≤ ϕ(c′′). Then, bythe symmetry of the argument, we conclude that ϕ(c′) = ϕ(c′′). This shows that ϕ is welldefined. Clearly, ϕ is the only map such that ϕ = ϕ ◦ ι ◦ β, so it remains to show that ϕis a PreW-morphism. It is clear that ϕ is a monoid homomorphism. Also, ϕ satisfies thecontinuity condition since ϕ does. Let c, d ∈ (B,�) such that c � d. We will show thatϕ(c) � ϕ(d). This is clear for d = 0, so assume that d > 0. Choose c′, d′ ∈ P such that(ι ◦ π)(c′) = c and (ι ◦ π)(d′) = d. We may find an element e′ ∈ P such that e′ ≺ d′ while also(ι◦β)(c′) = c < (ι◦β)(e′). It follows again that c′ ≤ e′ ≺ d′, thus ϕ(c) = ϕ(c′)� ϕ(d′) = ϕ(d).Finally, let c, d ∈ (B,�) such that c ≤ d. Let us prove that ϕ(c) ≤ ϕ(d). As (B,�) is a W-semigroup, we find a �-increasing sequence (cn)n in B with supremum c. By the previouspart, the sequence (ϕ(cn))n ⊆ S is increasing, and the inequality supn ϕ(cn) ≤ ϕ(c) holds. Lets ∈ S such that s � ϕ(c). By the continuity condition of ϕ, there exists an index k suchthat s ≤ ϕ(ck) ≤ supn ϕ(cn). Since s was arbitrary, we get ϕ(c) ≤ supn ϕ(cn). It follows thatϕ(c) = supn ϕ(cn) ≤ ϕ(d). This proves the claim.Overall, we have shown that there are bijections

HomCu(γ((Aff(K1)0++,�)⊗PreW (Aff(K2)0

++,�)), S)

∼=HomPreW((Aff(K1)0++,�)⊗PreW (Aff(K2)0

++,�), (S,�))

∼=HomPreW((B,�), (S,�))

∼=HomCu(γ(B,�), S)

which are natural in Cu-semigroups S. It follows that

γ((Aff(K1)0++,�)⊗PreW (Aff(K2)0

++,�)) ∼= γ(B,�),

as desired.

58


4.25 Theorem Let K1 and K2 be two metrizable compact convex sets. If either K1 or K2 isa Choquet simplex, there is a Cu-isomorphism


++∼= LAff(K1 ⊗K2)0

++.

Proof. We use Lemma 4.23 at the first step, Theorem 1.25 at the second step and Lemma 4.24at the third step to obtain


++∼= γ(Aff(K1)0

++,�)⊗ γ(Aff(K2)0++,�)

∼= γ((Aff(K1)0++,�)⊗PreW (Aff(K1)0

++,�))

∼= LAff(K1 ⊗K2)0++,

which is the desired result.

4.26 Corollary Let X and Y be metrizable compact Hausdorff spaces. There is a naturalCu-morphism

Lsc(X)0++ ⊗ Lsc(Y )0

++∼= Lsc(X × Y )0

++.

Proof. As X is a metrizable compact Hausdorff space, M+1 (X) is a metrizable Bauer simplex

and satisfies Lsc(X)0++∼= LAff(M+

1 (X))0++. An analogous statement holds for Y . According

to Corollary 4.13, we have that M+1 (X)⊗M+

1 (Y ) ∼= M+1 (X×Y ). We apply the main theorem

at the second step to deduce

Lsc(X)0++ ⊗ Lsc(Y )0

++∼= LAff(M+

1 (X))0++ ⊗ LAff(M+

1 (Y ))0++

∼= LAff(M+1 (X)⊗M+

1 (Y ))0++

∼= LAff(M+1 (X × Y ))0

++

∼= Lsc(X × Y )0++,

as desired.

59

5 The Cu-semigroup L(F (S))

In this chapter, our goal is to determine which Cu-semigroups are of the form LAff(K)0++, for

some metrizable compact convex setK. This problem requires us to come up with a meaningfulway to assign to each Cu-semigroup (with a few additional properties) a compact convex set. Inthe case of groups with order unit, the compact convex set consists of all normalized positivehomomorphisms into R. Similarly, in the case of Cu-semigroups, the compact convex setconsists of all functionals that are normalized at a certain element in the Cu-semigroup. Ourmain reference for this chapter is [Rob13].

5.1 Functionals

Recall that a functional on a Cu-semigroup S is a generalized Cu-morphism λ : S → [0,∞], i.e.a map that preserves the zero element, addition, order and suprema of increasing sequences.There are two trivial functionals on S: the functional λ0 that maps every element to zero andthe functional λ∞ that maps every nonzero element to ∞. We denote the set of functionalson S by F (S). When equipped with pointwise order and addition, F (S) becomes a positivelyordered monoid. Any upward directed subset of F (S) has a supremum, which is just givenby the pointwise supremum. Thus, F (S) is a dcpo, and the dcpo-structure is compatible withaddition. The formula (t ·λ)(s) := t ·λ(s) (for t ∈ (0,∞), λ ∈ F (S) and s ∈ S) defines a scalarmultiplication (0,∞) × F (S) → F (S). We extend this scalar multiplication to [0,∞] × F (S)

by setting

(0 · λ)(s) :=

0 if λ(s′) <∞ for all s′ � s

∞ otherwise, (∞ · λ)(s) :=

0 if λ(s) = 0

∞ otherwise,

for all λ ∈ F (S) and s ∈ S. Finally, we can endow F (S) with a certain topology that turnsF (S) into a compact Hausdorff space. This is done in the next Theorem.

5.1 Theorem ([ERS11, 4.8]) Let S be a Cu-semigroup. There exists a topology on F (S) inwhich a net (λi)i ⊆ F (S) converges to a functional λ ∈ F (S) if and only if for all s′, s ∈ Ssatisfying s′ � s, the inequality

lim supi

λi(s′) ≤ λ(s) ≤ lim inf

iλi(s)

60


holds. When equipped with this topology, F (S) becomes a compact Hausdorff space. If S iscountably based, then F (S) is second countable, hence metrizable.

5.2 Lemma Let S be a Cu-semigroup. Addition F (S) × F (S) → F (S) and scalar multipli-cation (0,∞)× F (S)→ F (S) are jointly continuous.

The proof of Lemma 5.2 is elementary and will be omitted. In [ERS11], it is erroneously statedthat the scalar multiplication [0,∞] × F (S) → F (S) is jointly continuous (where [0,∞] isequipped with the obvious topology that makes it homeomorphic to [0, 1]). For S = [0,∞]

and t ∈ [0,∞], let λt be the unique functional on S that maps 1 to t. It is easy to see thatthe sequence (λ 1

n) converges to λ0, yet ∞ · λ 1

n= λ∞ does not converge to ∞ · λ0 = λ0. This

counterexample was taken from [APT18, 5.2.1]. Similarly, the sequence (λn) converges to λ∞,but 0 · λn = λ0 does not converge to 0 · λ∞ = λ∞. Thus, scalar multilpication with 0 or with∞ is discontinuous.

We would like to study maps ϕ : F (S) → [0,∞] that preserve the structure of F (S). Possiblerequirements for ϕ could include:1) ϕ is a monoid homomorphism.2) ϕ is homogeneous for elements in (0,∞).3) ϕ is lower semicontinuous.4) ϕ is order-preserving.5) ϕ preserves suprema of upward directed subsets of F (S).In [Rob13], Robert considers maps ϕ : F (S)→ [0,∞] that satisfy the conditions 1), 2) and 3).He denotes the set of all these maps by Lsc(F (S)) (this is not to be confused with the definitionof Lsc(X) how it appears in this thesis). We will consider a different set: let LPoM(F (S))

denote the set of all lower semicontinuous PoM-morphisms ϕ : F (S) → [0,∞], i.e. all ϕ thatsatisfy the conditions 1), 3) and 4). We claim that such a ϕ automatically satisfies conditions 2)and 5). Let (λi)i ⊆ F (S) be an increasing net of functionals, and let us denote the supremumby λ. For all s′, s ∈ S with s′ � s, we have that

lim supi

λi(s′) = sup

iλi(s

′) = λ(s′) ≤ λ(s) = supiλi(s) = lim inf

iλi(s),

because the net is increasing. This shows that (λi)i converges to λ. Since ϕ is order-preserving,the net (ϕ(λi))i ⊆ [0,∞] is increasing, and ϕ(λ) is an upper bound. We use this at the secondand third step and the lower semicontinuity of ϕ at the first step to deduce

ϕ(λ) ≤ lim infi

ϕ(λi) = supiϕ(λi) ≤ ϕ(λ),

hence ϕ(λ) = supi ϕ(λi). This proves that ϕ satifies condition 5). Let µ ∈ F (S), t ∈ (0,∞).

61


It follows from 1) that ϕ(t · µ) = t · ϕ(µ) holds if t is rational. The general case follows usingcondition 5). Thus, ϕ also satisfies condition 2).This shows that our LPoM(F (S)) is contained in Robert’s Lsc(F (S)). It is proved in [Rob13,2.2.3] that F (S) is algebraically ordered if S satisfies (O5′) (the weak form of almost algebraicorder). In this case, every monoid homomorphism from F (S) into [0,∞] (or any other positivelyordered monoid) is automatically order-preserving. It follows that Robert’s Lsc(F (S)) agreeswith our LPoM(F (S)) if S satisfies (O5′).We equip LPoM(F (S)) with pointwise order and addition, giving it the structure of a pos-itively ordered monoid. Again, the pointwise supremum of any upward directed subset ofLPoM(F (S)) lies in LPoM(F (S)). Therefore LPoM(F (S)) is a dcpo, and the dcpo-structureis compatible with the addition. Also, we equip LPoM(F (S)) with the obvious scalar multi-plication (0,∞) × LPoM(F (S)) → LPoM(F (S)). This scalar multiplication satisfies all thecompatibility conditions you would hope for.Any element s ∈ S defines a map s : F (S) → [0,∞] by setting s(λ) := λ(s). It is easy tocheck that s lies in LPoM(F (S)) and that the canonical map S → LPoM(F (S)), s 7→ s isa PoM-morphism that preserves suprema of increasing sequences. If s is compact, then s iscontinuous.

5.2 The realification of a Cu-semigroup

It is natural to ask whether we can recover S from LPoM(F (S)) or more general from F (S). Theanswer to this question is no, because different Cu-semigroups may have the same functionals.Consider for example the Cu-semigroups N and [0,∞]. Any functional on N extends uniquely toa functional on [0,∞]. The underlying reason for this is that any functional λ : [0,∞]→ [0,∞]

satisfies λ(t) = t · λ(1) for any t ∈ (0,∞); hence λ is uniquely determined by the value λ(1).Thus, there are canonical bijections F ([0,∞]) ∼= F (N) ∼= [0,∞], and these bijections preserveall relevant structures.More general, if S is a Cu-semigroup and if there is a meaningful way to multiply elements inS with elements in (0,∞), the equation λ(t · s) = t · λ(s) holds for all t ∈ (0,∞), s ∈ S andλ ∈ F (S). This is captured in the upcoming definition. We can assign to each Cu-semigroup Sanother Cu-semigroup R(S) for which there is a meaningful way of multiplying with elementsin (0,∞) and which has a certain universal property. It will follow from this universal propertythat F (R(S)) ∼= F (S). We refer to R(S) as the realification of S. In the example above, [0,∞]

is the realification of N. The realification of S turns out to be a useful tool for studying thenatural map S → LPoM(F (S)).The following definition is a special case of [APT18, 7.1.3].

62


5.3 Definition and Proposition Let S be a Cu-semigroup. A [0,∞]-multiplication on S is aCu-bimorphism m : [0,∞]× S → S such that the equations

m(t1,m(t2, s)) = m(t1 · t2, s) and m(1, s) = s

hold for all t1, t2 ∈ [0,∞] and s ∈ S. A [0,∞]-multiplication is unique: if m and m′ are two[0,∞]-multiplications on S, then m = m′. Moreover, for any n ∈ N and s ∈ S, we have thatm(n, s) = n ·s. It is therefore justified to write t ·s instead of m(t, s) (for t ∈ [0,∞] and s ∈ S).A Cu-semigroup S is said to have [0,∞]-multiplication if there exists a [0,∞]-multiplicationon S. Since the [0,∞]-multiplication on S is unique, we will regard it as a property of S ratherthan part of the data.Let S and T be two Cu-semigroups with [0,∞]-multiplication. Any generalized Cu-morphismϕ : S → T preserves the [0,∞]-multiplications, i.e. ϕ(t · s) = t · ϕ(s) holds for any t ∈ [0,∞]

and s ∈ S.

Proof. For i = 1, 2, suppose thatmi is a real multiplication on a Cu-semigroup Si. Additionally,let ϕ : S1 → S2 be a generalized Cu-morphism. Let s ∈ S1. For any n ∈ N, we have that

m1(n, s) = m1(

n∑i=1

1, s) =

n∑i=1

m1(1, s) =

n∑i=1

s = n · s

and similarly m2(n, ϕ(s)) = n · ϕ(s), hence

ϕ(m1(n, s))) = ϕ(n · s) = n · ϕ(s) = m2(n, ϕ(s)).

Now let n ∈ Z and k ∈ N∗. The computation

ϕ(m1(k−1n, s)) = m2(k−1,m2(k, ϕ(m1(k−1n, s)))) = m2(k−1, ϕ(m1(k,m1(k−1n, s))))

= m2(k−1, ϕ(m1(n, s))) = m2(k−1,m2(n, ϕ(s))) = m2(k−1n, ϕ(s))

shows that the formula ϕ(m1(q, s)) = m2(q, ϕ(s)) holds for every rational number q. Finally,let t ∈ R. Choose an increasing sequence (qn)n∈N of rational numbers with supremum t. Wehave that

ϕ(m1(t, s)) = ϕ(supnm1(qn, s)) = sup

nϕ(m1(qn, s)) = sup

nm2(qn, ϕ(s)) = m2(t, ϕ(s)),

proving that ϕ preserves the [0,∞]-multiplications.In the special case of S1 = S2 = S and ϕ = id, this shows that the real multiplication on S isunique.

5.4 Example The Cu-semigroup [0,∞] has [0,∞]-multiplication, where multiplication with0 and ∞ is defined by 0 · s := 0, for all s ∈ [0,∞], and ∞ · s =∞, for all nonzero s.If K is a metrizable compact convex set, then LAff(K)0

++ has [0,∞]-multiplication, by setting(t · f)(x) := t · f(x), for all t ∈ [0,∞], f ∈ LAff(K)0

++ and x ∈ K.

63


5.5 Theorem For every Cu-semigroup S, there exists a Cu-semigroup R(S) with [0,∞]-multiplication and a generalized Cu-morphism γ : S → R(S) such that the following universalproperty holds:For any Cu-semigroup T that has [0,∞]-multiplication and for any generalized Cu-morphismϕ : S → T , there exists a unique generalized Cu-morphism ϕ : R(S)→ T satisfying ϕ = ϕ ◦ γ.If ϕ is a Cu-morphism, then ϕ is a Cu-morphism as well.As usual, R(S) is uniquely determined up to Cu-isomorphism. In fact, (R(S), γ) can be realizedas [0,∞] ⊗ S and γ : S → [0,∞] ⊗ S, γ(s) = 1 ⊗ s. Thus, S has [0,∞]-multiplication if andonly if S ∼= [0,∞]⊗ S. We refer to (R(S), γ) as the realification of S.

Proof. Set R(S) := [0,∞] ⊗ S, and define γ : S → R(S) by the formula γ(s) := 1 ⊗ s. It isclear that γ is a generalized Cu-morphism. To show that R(S) has [0,∞]-multiplication, lett ∈ [0,∞]. The map [0,∞]× S → R(S), (α, s) 7→ (t · α)⊗ s is a generalized Cu-bimorphism.Let us denote the corresponding generalized Cu-morphism R(S) → R(S) by mt. Then wedefine m : [0,∞]×R(S)→ R(S) by setting m(t, x) := mt(x). If α⊗ s ∈ R(S) is an elementarytensor, we have that m(t, α⊗ s) = (t · α)⊗ s. It is fairly straightforward, but tedious, to showthat m is indeed a [0,∞]-multiplication on R(S). We will omit the argument.Let T be a Cu-semigroup with [0,∞]-multiplication, and let ϕ : S → T be a generalized Cu-morphism. The map [0,∞] × S → T , which is defined by (α, s) 7→ α · ϕ(s), is a generalizedCu-bimorphism. The corresponding generalized Cu-morphism ϕ : R(S) → T clearly satisfiesϕ = ϕ ◦ γ, so only uniqueness remains to be shown. Let ψ : R(S)→ T be another generalizedCu-morphism satisfying ϕ = ψ ◦ γ. Using that any generalized Cu-morphism preserves [0,∞]-multiplication at the second step, we deduce

ψ(α⊗ s) = ψ(α · (1⊗ s)) = α · ψ(1⊗ s) = α · ϕ(s) = ϕ(α⊗ s),

for any elementary tensor α⊗ s ∈ R(S), implying that ψ = ϕ.

Since the Cu-semigroup [0,∞] has [0,∞]-multiplication, it follows from the universal propertyof the realification of a Cu-semigroup S that there exists a natural bijection F (S) ∼= F (R(S)).This bijection is compatible with all the relevant structures (this is easy, albeit not entirely triv-ial). The induced map LPoM(F (S)) ∼= LPoM(F (R(S))) is also a bijection that is compatiblewith all the structures. The diagram

S LPoM(F (S))

R(S) LPoM(F (R(S)))

∼=

is commutative. From now on, we will not differentiate between F (S) and F (R(S)), between

64


LPoM(F (S)) and LPoM(F (R(S))) and between the maps S → LPoM(F (S)) and R(S) →LPoM(F (S)).

5.3 The range of the natural map

Given two maps f, g : F (S) → [0,∞], we write f C g if there exists some ε > 0 such thatf ≤ (1 − ε)g and if f is continuous at all points where g is finite. In [Rob13], Robert usesL(F (S)) to denote the set of all f ∈ Lsc(F (S)) for which there exists a C-increasing sequencein Lsc(F (S)) with supremum f (where Lsc(F (S)) is defined as in [Rob13]). In this thesis, weuse L(F (S)) to denote the set of all f ∈ LPoM(F (S)) for which there exists a C-increasingsequence in LPoM(F (S)) with supremum f . Again, in general our L(F (S)) is contained inRobert’s L(F (S)), and both agree if S satisfies (O5’).If f C g, then f � g, where � denotes the sequential way-below relation in LPoM(F (S)).We refer to [ERS11, 5.1]. An adapted proof shows that this is even true for the non-sequentialway-below relation. One can show that L(F (S)) is a submonoid of LPoM(F (S)) that is closedunder scalar multiplication and suprema of increasing sequences.The bijection LPoM(F (S)) ∼= LPoM(F (R(S))) is compatible with the C-relations, so it re-stricts to a bijection L(F (S)) ∼= L(F (R(S))). Thus, we will not differentiate between L(F (S))

and L(F (R(S))).Lemma 5.6 and Proposition 5.7 are adapted from [Rob13, 3.1.6].

5.6 Lemma Let S be a Cu-semigroup with [0,∞]-multiplication. If a, c ∈ S satisfy a � c,then there exists b ∈ S such that a ≤ b� c while also a ≤ b C c.

Proof. Choose some element d ∈ S and ε > 0 such that a � d � (1 − ε)c ≤ c. Then, byinduction, we may choose elements sr ∈ S, where r runs through the dyadic rationals in [0, 1],such that s0 = a, s1 = d and sr � sr′ whenever r < r′. For n ∈ N∗, set

bn :=1

2n·

2n−1∑k=0

s k2n

and bn :=1

2n·

2n∑k=1

s k2n.

Although not entirely obvious, it is easy to check that the following conditions are satisfied:• The sequence (bn)n ⊆ S is �-increasing.• The sequence (bn)n ⊆ S is �-decreasing.• For every n, the inequality a ≤ bn ≤ bn ≤ d holds.

Let b := supn bn ∈ S. Then a ≤ b ≤ d� (1− ε)c ≤ c, whence a ≤ b� c and a ≤ b ≤ (1− ε)c.It remains to show that b is continuous at each point where c is finite. Let λ ∈ F (S) such thatc(λ) < ∞, and let (λi)i ⊆ F (S) be a net that converges to λ. For any m ∈ N∗, we have that

65


bm � b, hencelim sup

ibm(λi) = lim sup

iλi(bm) ≤ λ(b) = b(λ)

by the description of the convergence in F (S). As d� c, we obtain lim supi d(λi) ≤ c(λ) withthe same reasoning. Moreover, we have that b ≤ bm ≤ bm + d

2m (we used that the sequence(bn)n is decreasing at the first step), hence b ≤ bm + 1

2m d. We obtain that

lim supi

b(λi) ≤ (lim supi

bm(λi)) +1

2m(lim sup

id(λi)) ≤ b(λ) +

c(λ)

2m.

Since c(λ) is finite and m is arbitrary, we conclude that lim supi b(λi) ≤ b(λ). This shows thatb is upper semicontinuous (and hence continuous) in λ. Therefore a ≤ b C c, as desired.

5.7 Proposition Let s be an element in some Cu-semigroup S. There exists a sequence(sn)n ⊆ R(S) such that the sequence (sn)n ⊆ LPoM(F (S)) is C-increasing with supremum s.The sequence (sn)n may be chosen to be �-increasing. In particular, s lies in L(F (S)).

Proof. We may assume without loss of generality that S has [0,∞]-multiplication. Choose a�-increasing sequence (tn)n ⊆ S with supremum s. For each n, apply Lemma 5.6 to obtainan element sn ∈ S such that tn ≤ sn � tn+1 while also tn ≤ sn C tn+1. The sequence (sn)n is�-increasing with supremum s, and the sequence (sn)n is C-increasing with supremum s.

5.8 Theorem ([Rob13, 3.2.1]) Let S be a Cu-semigroup satisfying (O5’). Then L(F (S)) is aCu-semigroup, and the natural map R(S)→ L(F (S)) is a Cu-isomorphism.

Proof. In [Rob13], the Cu-semigroup S is always assumed to satisfy (O5’). Remember that inthis case, there is no difference between Robert’s Lsc(F (S)) and our LPoM(F (S)) and betweenRobert’s L(F (S)) and our L(F (S)). Robert defines SR to be the subset of LPoM(F (S)) thatconsists of suprema of increasing sequences coming from the set { sn | n ∈ N∗, s ∈ S}. He showsthat SR is a Cu-semigroup and that SR = L(F (S)). It is shown in [APT18, 7.5.9] that thenatural map R(S)→ SR is a Cu-isomorphism.

5.4 The image of the functor LAff(_)0++

For a fixed element u ∈ S, let Fu(S) denote the set of all functionals on S that are normalizedat u, that is all functionals λ ∈ F (S) such that λ(u) = 1. Then Fu(S) is a convex subset ofF (S) in the sense that t · λ+ (1− t) · µ lies in Fu(S) for all λ, µ ∈ Fu(S) and t ∈ [0, 1]. Noticethat Fu(S) is a closed (and therefore compact) subset of F (S) if u is continuous. The followinglemma demonstrates that we can recover F (S) from Fu(S) if S is simple.

66


5.9 Lemma Let S be a simple Cu-semigroup, and let u ∈ S \ {0} such that u is continuous.Then F (S) = ((0,∞) · Fu(S)) ∪ {λ0, λ∞}.

Proof. Let λ ∈ F (S). Suppose first that λ(u) = 0. Let s ∈ S. Every s′ � s is way-below∞. Since u is nonzero and S is simple, there exists n ∈ N such that s′ ≤ nu, implying thatλ(s′) = 0. Passing to the supremum yields λ(s) = 0, whence λ = λ0. If λ(u) ∈ (0,∞), thenλ

λ(u) lies in Fu(S), so λ lies in (0,∞) · Fu(S). Finally, suppose that λ(u) =∞. Let us assumethat λ 6= λ∞. This implies that 0 · λ = λ0, because S is simple. The sequence (λn)n convergesto λ0. Since u is continuous, we get that

0 = u(λ0) = limnu(λ

n) = lim

n

λ(u)

n=∞,

which is a contradiction. Therefore λ = λ∞.

The following lemma is adapted from [Rob13, 3.2.3].

5.10 Lemma Let S be a simple and nonzero Cu-semigroup. Set C := F (S) \ {λ∞}, and letV denote the set of all maps C → [0,∞) that are additive, continuous and homogeneous forelements in [0,∞). A net (λi)i ⊆ C converges to a functional λ ∈ C if and only if (ϕ(λi))i

converges to ϕ(λ) for all ϕ ∈ V .

Proof. In order to prove the nontrivial implication, let a, b ∈ S satisfy a� b. We have to showthat the inequalities

lim supi

λi(a) ≤ λ(b) ≤ lim infi

λi(b)

hold. Choose c ∈ S such that a � c � b, and let ε > 0. It is shown in [Rob13, 2.2.5] that(1−ε)c� b. Using Proposition 5.7, we may choose d, e ∈ R(S) such that (1−ε)c� d C e� b.We claim that e is finite on C. To see this, choose x ∈ S such that x� b and such that e� x

(this is possible since b = supy�b y). If λ lies in C, we may choose an element s ∈ S \ {0}such that λ(s) <∞. Since S is simple and since x�∞, there exists n ∈ N such that x ≤ ns,implying that x(λ) <∞. Therefore x is finite on C, and the same can be said for e.We conclude that the restriction of d to C lies in V . Using our assumption at the second stepyields

(1− ε) lim supi

a(λi) ≤ lim supi

d(λi) = d(λ) ≤ b(λ).

As ε may be arbitrary small, we obtain lim supi λi(a) ≤ λ(b). Similarly, we have that

(1− ε)c(λ) ≤ d(λ) = lim infi

d(λi) ≤ lim infi

b(λi),

and therefore λ(c) ≤ lim infi λi(b) by passing to the supremum of all ε > 0. Since c � b

is arbitrary, and since λ preserves suprema of increasing sequences, we deduce that λ(b) ≤lim infi λi(b). Overall, we have shown that (λi)i converges to λ, as desired.

67


5.11 Proposition Let S be a simple Cu-semigroup, and let u ∈ S such that u is continuous.Then Fu(S) is a compact convex subset of a locally convex, Hausdorff, real topological vectorspace. If S is countably based, then Fu(S) is metrizable. If S satisfies (O5′) and (O6), thenFu(S) is a Choquet simplex.

Proof. The statements are clear if u = 0, so let us assume that u 6= 0. Suppose that λ, µ, η inC := F (S) \ {λ∞} satisfy λ+ η = µ+ η. Then λ and µ agree on the ideal {s ∈ S | η(s) <∞}.Since η 6= λ∞ and since S is simple, this ideal is equal to S, so λ = µ. Therefore C is acancellative monoid, so C can be understood as a submonoid of its Grothendieck completion,which we denote by E. We endow E with the obvious scalar multiplication, giving it thestructure of a real vector space. Let V be the set of all maps C → [0,∞) that are additive,continuous and homogeneous for elements in [0,∞). For each ϕ ∈ V , we define a seminormon E by setting ‖λ − µ‖ϕ := |ϕ(λ) − ϕ(µ)|. We equip E with the topology induced by theseseminorms, giving E the structure of a locally convex real topological vector space. By Lemma5.10, the topology of C as a subset of F (S) agrees with the topology of C as a subset of E.Since C is hausdorff, E is a Hausdorff space as well. Therefore, we can regard Fu(S) as a subsetof the locally convex, Hausdorff, real topological vector space E. Also, Fu(S) is contained ina hyperplane which misses the origin, so it follows from Lemma 5.9 that the cone with baseFu(S) is given by C.If S is countably based, then F (S) is metrizable according to Theorem 5.1, so Fu(S) is metriz-able as well. Under the assumption that S satisfies (O5′) and (O6), it is shown in [Rob13, 4.1.2]that F (S) is a lattice. Then C must be a lattice as well, so Fu(S) is a Choquet simplex.

5.12 Proposition Suppose that S is a countably based, simple Cu-semigroup satisfying (O5′).Let u ∈ S \ {0} such that u is continuous. The restriction map

ρ : L(F (S))→ LAff(Fu(S))0++

is a Cu-isomorphism.

Proof. It is clear that the restriction of an element in L(F (S)) to Fu(S) is lower semicontinuousand affine. However, it is not immediately clear that such a restriction is strictly positive (orzero). According to Theorem 5.8, every element in L(F (S)) has the form a for some a ∈ R(S).Let us suppose that a 6= 0. We claim that ρ(a) > 0. To see this, we may choose s ∈ S \{0} andn ∈ N∗ such that s

n ≤ a, implying that 1nρ(s) ≤ ρ(a). Thus, it suffices to show that ρ(s) > 0.

Let λ ∈ Fu(S) and choose u′ � u such that λ(u′) > 0. Since S is simple and s is nonzero,there exists m ∈ N such that u′ ≤ m · s. It follows that

0 < λ(u′) ≤ m · λ(s) = m · ρ(s)(λ),

68


whence ρ(s) > 0, as desired.Let us show that any element f ∈ LAff(Fu(S))0

++ extends uniquely to some element f inL(F (S)). We have seen in the proof of Proposition 5.11 that the cone with base Fu(S) is givenby F (S) \ {λ∞} (under the assumption that u 6= 0). Therefore f extends uniquely to a lowersemicontinuous PoM-morphism f : F (S) \ {λ∞} → [0,∞]. We extend f to F (S) by settingf(λ∞) := 0 if f = 0 and f(λ∞) := ∞ if f > 0. It is clear that f lies in LPoM(F (S)), soit remains to show that f actually lies in L(F (S)). Choose a �-increasing sequence (fn)n inAff(Fu(S))0

++ with supremum f . Let n ∈ N. Note that fn � fn+1 entails that there existssome ε > 0 such that fn ≤ (1 − ε)fn+1, whence fn ≤ (1 − ε)fn+1. Since fn is continuous,the extension onto F (S) \ {λ∞} is continuous as well. It follows that the sequence (fn)n isC-increasing with supremum f . In particular, f lies in L(F (S)).We have seen that ρ is a PoM-isomorphism. In general, any PoM-isomorphism of Cu-semigroupsis automatically a Cu-isomorphism. Thus, ρ is a Cu-isomorphism.

We know already that we can recover any compact convex set K from Aff(K) by applyingthe state functor. We will see in the upcoming Proposition that we can also recover K fromLAff(K)0

++ by taking normalized functionals.

5.13 Proposition Let K be a metrizable compact convex set and consider 1 ∈ LAff(K)0++.

Then 1 is continuous. The natural map

κ : K → F1(LAff(K)0++), x 7→ evx

is an affine homeomorphism.

Proof. Suppose that (λi)i ⊆ F (LAff(K)0++) is a net that converges to some functional λ in

F (LAff(K)0++). All t ∈ (0, 1) satisfy t� 1 (in LAff(K)0

++), so the inequality

t · lim supi

λi(1) = lim supi

λi(t) ≤ λ(1)

holds. Passing to the supremum yields lim supi 1(λi) ≤ 1(λ). We have shown that 1 is uppersemicontinuous, hence continuous.It is clear that κ is affine. The injectivity follows from the Hahn-Banach theorem. In order toshow that κ is continuous, let (xi)i ⊆ K be a net that converges to some point x ∈ K. Letf, h ∈ LAff(K)0

++ satisfy f � h. Choose g ∈ Aff(K) such that f ≤ g � h. If we use thecontinuity of g at the second and the lower semicontinuity of h at the fourth step, we obtain

lim supi

f(xi) ≤ lim supi

g(xi) = g(x) ≤ h(x) ≤ lim infi

h(xi),

so κ(xi) converges to κ(x), as desired.

69


For the surjectivity, let λ ∈ F1(LAff(K)0++). Notice that the restriction of λ to Aff(K)0

++ isreal-valued (because 1 is an order unit in Aff(K)0

++). The function ϕ := Grot(λ|Aff(K)0++) lies

in S(Aff(K), <=, 1) = S(Aff(K),≤, 1). By Proposition 2.11, there exists an element x ∈ K

such that ϕ = evx. But then κ(x) = λ (since they agree on the dense subset Aff(K)0++), so κ

is indeed surjective.

5.14 Corollary A metrizable compact convex set K is a Choquet simplex if and only ifLAff(K)0

++ satisfies (O6).

Proof. If K is a Choquet simplex, then LAff(K) is an inf-semilattice according to Proposition3.12. It follows easily from this that LAff(K)0

++ satisfies (O6). The argument is omitted.Conversely, assume that LAff(K)0

++ satisfies (O6). It then follows from Propositions 5.13 and5.11 that K ∼= F1(LAff(K)0

++) is a Choquet simplex.

Recall that a Cu-semigroup S is called almost unperforated if for all x, y ∈ S, the condition(k + 1)x ≤ ky for some k ∈ N implies that x ≤ y. We say that S is almost divisible if for allk ∈ N and x, y ∈ S satisfying x� y, there exists z ∈ S such that x ≤ (k + 1)z and kz ≤ y.

5.15 Theorem ([APT18, 7.5.4.]) A Cu-semigroup S has [0,∞]-multiplication if and only if Sis almost unperforated, almost divisible, and every element of S is soft.

5.16 Theorem For a Cu-semigroup S, the following are equivalent:1) There exists a metrizable compact convex set K such that S ∼= LAff(K)0

++.2) S is simple, countably based and satisfies (O5′). Moreover, S is almost unperforated, almost

divisible, and every element of S is soft.

Proof. It is clear that 1) implies 2). Now assume that 2) holds. If S = 0, we can chooseK = ∅. Now assume that S 6= 0. Then we can choose a, b ∈ S such that 0 6= a � b � ∞.By Theorem 5.15, S has [0,∞]-multiplication. Lemma 5.6 allows us to find an element u ∈ Ssuch that a ≤ u � b while also a ≤ u C b. Clearly, u is nonzero. Since we have chosen b tobe way below ∞, and since S is simple, b is finite on F (S) \ {λ∞}. Therefore u is continuouson F (S) \ {λ∞}. As u is automatically continuous in λ∞, we have that u is continuous. ThenK := Fu(S) is a metrizable compact convex set according to Proposition 5.11. We use thatS has [0,∞]-multiplication at the first step, Theorem 5.8 at the second step and Proposition5.12 at the third step to deduce

S ∼= R(S) ∼= L(F (S)) ∼= LAff(K)0++,

which is the desired result.

70


Remark It follows from Theorem 5.16 and the main theorem that the full subcategory of Cu,consisting of simple, countably based, almost unperforated, almost divisible Cu-semigroupssatisfying (O5′) and (O6) and which consist only of soft elements is closed under the tensorproduct in Cu.

71

6 An application of the main theorem

One of the main motivations for studying LAff(K)0++ is that this semigroup appears naturally

as the soft part of Cuntz semigroups coming from sufficiently nice C∗-algebras. In this chapter,we use the main theorem to relate Cu(A) ⊗ Cu(B) to Cu(A ⊗min B), for C∗-algebras A andB. This is merely meant to be an outlook on what could be done with the main theorem, sowe will not provide any proofs. In fact, the argumentation is speculative at times. The readershould keep that in mind.

If A is any unital C∗-algebra, then both the set of all normalized traces T1(A) and the set ofall normalized quasitraces QT1(A) can be regarded as Choquet simplices (see [Sak71, 3.1.18]and [BH82, II.4.4]). If A is separable, then T1(A) and QT1(A) are metrizable. Now supposethat A and B are unital, simple, separable, stably finite, Z-stable C∗-algebras. By Proposition1.32, their Cuntz semigroups can be computed as

Cu(A) ∼= V (A) tγA LAff(QT1(A))0++, and

Cu(B) ∼= V (B) tγB LAff(QT1(B))0++,

where γA and γB are the obvious composition maps. Since all those properties pass to theminimal tensor product, we also have that

Cu(A⊗min B) ∼= V (A⊗min B) tγ LAff(QT1(A⊗min B))0++,

with the obvious composition map γ. It follows from the main theorem and the fact that thetensor product of two Choquet simplices is again a Choquet simplex that

LAff(QT1(A))0++ ⊗ LAff(QT1(B))0

++∼= LAff(QT1(A)⊗QT1(B))0

++

is a simple semigroup in the category D (see below Definition 1.27 on page 14). Let us assumethat V (A) ⊗PoM V (B) is a simple semigroup in the category C, and that the canonical PoM-morphism

γA ⊗ γB : V (A)⊗PoM V (B)→ LAff(QT1(A))0++ ⊗ LAff(QT1(B))0

++

is a composition map. We can then compose these semigroups to obtain a Cu-semigroup

(V (A)⊗PoM V (B)) tγA⊗γB (LAff(QT1(A))0++ ⊗ LAff(QT1(B))0

++),

72


and it is hopefully not too far-fetched to assume that

Cu(A)⊗ Cu(B) ∼= (V (A)⊗PoM V (B)) tγA⊗γB (LAff(QT1(A))0++ ⊗ LAff(QT1(B))0

++).

For arbitrary C∗-algebras A and B, there is a natural Cu-morphism Cu(A) ⊗ Cu(B) →Cu(A ⊗min B) that factors through Cu(A ⊗max B). This Cu-morphism is known to be aCu-isomorphism if one of the C∗-algebras is an AF-algebra (cf. [APT18, 6.4.13]). In general,however, it is not an isomorphism. Any Cu-morphism maps compact elements to compactelements and soft elements to soft elements. In our case, we obtain a Cu-morphism


++ → LAff(QT1(A⊗min B))0++.

In the case that Cu(A)⊗ Cu(B)→ Cu(A⊗min B) is a Cu-isomorphism, it follows that

LAff(QT1(A)⊗QT1(B))0++∼= LAff(QT1(A))0

++ ⊗ LAff(QT1(B))0++

∼= LAff(QT1(A⊗min B))0++.

We can then apply Proposition 5.13 to deduce QT1(A ⊗min B) ∼= QT1(A) ⊗ QT1(B). To theauthor’s knowledge, there is no complete description of QT1(A⊗minB) for general C∗-algebrasA and B. On the other hand, there is this result for traces, which is due to Guichardet [Gui69,Proposition 22]: for all unital C∗-algebras A and B, we have that

∂eT1(A⊗min B) = {τA ⊗ τB | τA ∈ ∂eT1(A), τB ∈ ∂eT1(B)}.

In view of Theorem 4.9, this suggests that T1(A⊗min B) ∼= T1(A)⊗T1(B) holds for all unitalC∗-algebras A and B. Haagerup proved that every quasitrace on a unital, exact C∗-algebra isa trace [Haa14]. Again, this suggests that QT1(A ⊗min B) ∼= QT1(A) ⊗ QT1(B) holds for allunital, exact C∗-algebras.We return to our original assumptions on A and B. Let us assume that there is a naturalisomorphism QT1(A⊗min B) ∼= QT1(A)⊗QT1(B). We obtain a natural Cu-isomorphism


++∼= LAff(QT1(A)⊗QT1(B))0

++

∼=−→ LAff(QT1(A⊗min B))0++.

There is a canonical PoM-morphism V (A) ⊗PoM V (B) → V (A ⊗min B) that factors throughV (A ⊗max B). If we combine these two maps, we obtain a Cu-morphism ω such that thediagram

Cu(A⊗min B) V (A⊗min B) tγ LAff(QT1(A⊗min B))0++

Cu(A)⊗ Cu(B) (V (A)⊗PoM V (B)) tγA⊗γB (LAff(QT1(A))0++ ⊗ LAff(QT1(B))0

++)

∼=

∼=

ω

73


commutes. In the case that V (A)⊗PoMV (B)→ V (A⊗minB) is an isomorphism, then ω shouldbe an isomorphism as well. Consequently, the natural map Cu(A)⊗ Cu(B) → Cu(A⊗min B)

is also an isomorphism. Thus, we should be able to derive a criterion for which the naturalCu-morphism Cu(A)⊗ Cu(B)→ Cu(A⊗min B) is a Cu-isomorphism.

74

Bibliography

[Alf71] E. M. Alfsen. Compact convex sets and boundary integrals. Springer-Verlag, NewYork-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band57.

[APT18] R. Antoine, F. Perera, and H. Thiel. Tensor products and regularity properties ofCuntz semigroups. Mem. Amer. Math. Soc., 251(1199):viii+191, 2018.

[Arv96] W. Arveson. Notes on measure and integration in locally compact spaces, 1996.Available from https://math.berkeley.edu/~arveson/Dvi/rieszMarkov.pdf.

[BH82] Bruce Blackadar and David Handelman. Dimension functions and traces on C∗-algebras. J. Funct. Anal., 45(3):297–340, 1982.

[CEI08] K. T. Coward, G. A. Elliott, and C. Ivanescu. The Cuntz semigroup as an invariantfor C∗-algebras. J. Reine Angew. Math., 623:161–193, 2008.

[Cun78] J. Cuntz. Dimension functions on simple C∗-algebras. Math. Ann., 233(2):145–153,1978.

[Edw65] David Albert Edwards. Séparation des fonctions réelles définies sur un simplexe deChoquet. C. R. Acad. Sci. Paris, 261:2798–2800, 1965.

[EHS80] Edward G. Effros, David E. Handelman, and Chao Liang Shen. Dimension groupsand their affine representations. Amer. J. Math., 102(2):385–407, 1980.

[Eng14] M. Engbers. Decomposition of simple Cuntz semigroups. PhD thesis, Westfälis-che Wilhelms-Universität Münster, 2014. Available from https://core.ac.uk/download/pdf/56470722.pdf.

[ERS11] G. A. Elliott, L. Robert, and L. Santiago. The cone of lower semicontinuous traceson a C∗-algebra. Amer. J. Math., 133(4):969–1005, 2011.

[Fre72] D. H. Fremlin. Tensor products of Archimedean vector lattices. Amer. J. Math.,94:777–798, 1972.

[GH86] K. R. Goodearl and D. E. Handelman. Tensor products of dimension groups and K0

of unit-regular rings. Canad. J. Math., 38(3):633–658, 1986.[Goo86] K. R. Goodearl. Partially ordered abelian groups with interpolation, volume 20 of

Mathematical Surveys and Monographs. American Mathematical Society, Providence,RI, 1986.

[Gui69] A. Guichardet. Tensor products of C∗-algebras, 1969. Available from http://www.

75

https://math.berkeley.edu/~arveson/Dvi/rieszMarkov.pdf

https://core.ac.uk/download/pdf/56470722.pdf

https://core.ac.uk/download/pdf/56470722.pdf

http://www.fuw.edu.pl/~kostecki/scans/guichardet1969.pdf


Bibliography

fuw.edu.pl/~kostecki/scans/guichardet1969.pdf.[Haa14] U. Haagerup. Quasitraces on exact C∗-algebras are traces. C. R. Math. Acad. Sci.

Soc. R. Can., 36(2-3):67–92, 2014.[Jel68] Francis Jellett. Homomorphisms and inverse limits of Choquet simplexes. Math. Z.,

103:219–226, 1968.[LOS78] J. Lindenstrauss, G. Olsen, and Y. Sternfeld. The Poulsen simplex. Ann. Inst. Fourier

(Grenoble), 28(1):vi, 91–114, 1978.[NP69] I. Namioka and R. R. Phelps. Tensor products of compact convex sets. Pacific J.

Math., 31:469–480, 1969.[Rob13] L. Robert. The cone of functionals on the Cuntz semigroup. Math. Scand., 113(2):161–

186, 2013.[RW10] M. Rørdam and W. Winter. The Jiang-Su algebra revisited. J. Reine Angew. Math.,

642:129–155, 2010.[Sak71] Shôichirô Sakai. C∗-algebras and W ∗-algebras. Springer-Verlag, New York-

Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60.[Thi16] H. Thiel. The Cuntz semigroup. Lecture notes, 2016. Available from https://

ivv5hpp.uni-muenster.de/u/h_thie08/teaching/CuScript.pdf.[Tom08] A. S. Toms. On the classification problem for nuclear C∗-algebras. Ann. of Math.

(2), 167(3):1029–1044, 2008.[Weh96] F. Wehrung. Tensor products of structures with interpolation. Pacific J. Math.,

176(1):267–285, 1996.

76



https://ivv5hpp.uni-muenster.de/u/h_thie08/teaching/CuScript.pdf

https://ivv5hpp.uni-muenster.de/u/h_thie08/teaching/CuScript.pdf

tensor products of cuntz semigroups associated with

Documents