lonke msc thesis

Extensions of pure states and the Laurent operator

Yossi Lonke

M.Sc Thesis1

1Submitted to the Hebrew University at Jerusalem, 1991

2

This thesis has been prepared under the direction of Prof. Lior Tzafriri,to whom I am greatly indebted.

Contents

1 Introduction 51.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The Kadison-Singer Problem . . . . . . . . . . . . . . . . . . 61.3 Pure states in Quantum Mechanics . . . . . . . . . . . . . . . 71.4 Pure states and representation theory . . . . . . . . . . . . . 9

2 The pure states of D 132.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The pure states of the algebra l∞(N) . . . . . . . . . . . . . . 142.3 Extensions of pure states of D . . . . . . . . . . . . . . . . . . 162.4 Glimm’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . 19

3 The Laurent Operator 233.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 The family H(ϕ, ε) . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 The paving property and syndetic sets . . . . . . . . . 29

4 Equivalent Formulations 334.1 Six equivalent formulations . . . . . . . . . . . . . . . . . . . 334.2 uniqueness of extension ⇐⇒ diagonalization property . . . . 354.3 uniqueness of extension =⇒ paving property . . . . . . . . . . 354.4 paving property =⇒ relative Dixmier property . . . . . . . . 354.5 paving property ⇐⇒ local paving property . . . . . . . . . . 364.6 local paving property =⇒ restricted invertibility . . . . . . . 374.7 restricted invertibility =⇒ local paving property . . . . . . . 38

3

4 CONTENTS

Chapter 1

Introduction

1.1 Overview

Chapter 1 introduces the main concepts and notation. In chapter 2 thereis nothing new; we have brought together some known characterizations ofthe pure states P (D) of D: P (D) is the Stone-Cech compactification of thenatural numbers N, and there exists a one-to-one correspondence betweenmembers of P (D) and ultra-filters over N. If D is the diagonal algebra ofB(l2(Z)), then everything in chapter 2 holds with N replaced by Z.

In 2.3 we make use of an argument of Kadison and Singer to give anecessary and sufficient condition for a pure state f ∈ P (D) to have aunique extension in terms of the ultra-filter Uf that corresponds to f . Foreach f ∈ P (D) and T ∈ B(l2(N)) we define:

βf (T ) = inf||Pσ(T − E(T ))Pσ|| : σ ∈ Uf

and show that βf (T ) = 0 for every T ∈ B(l2(N)) if and only if f has a uniqueextension to a pure state of B(l2(N)). A similar result holds for B(l2(Z)).As a corollary we will have proved that the non-singular pure states of Ddo have a unique extension to pure states of B(l2(N)), a fact that is knownalready.

In 2.4 we give a modest partial answer to a conjecture of Glimm, whoin [9] conjectured that if A is a proper C∗ sub-algebra of of C∗ algebra B,then there exist distinct pure states on B whose restriction to A coincide. Inother words, C∗ sub-algebras do not separate the pure states of C∗-algebrasthat strictly contain them. We will show that Glimm’s conjecture holds inthe special case when A is abelian. This however does not provide a negativeanswer to the KS problem, because there are always pure states on the largeralgebra whose restriction to the smaller one is not a pure state.

The third chapter deals with the Laurent operator. The two main resultsare theorems 3.2.1, 3.2.3. The first claims that if ϕ ∈ L∞(0, 2π) is real,and ε > 0, and H(ϕ, ε) is the family of sets Λ ⊂ Z for which ||PΛ(Lϕ −

5

6 CHAPTER 1. INTRODUCTION

ϕ(0)I)PΛ|| ≤ ε then H(ϕ, ε) contains a set Λ ⊂ Z of positive density, that is

limN→∞|Λ∩[1,N ]

N > 0, and the second charaterizes those Laurent operatorshaving the paving property: Lϕ has the paving property if and only ifH(ϕ, ε)contains a syndetic subset of Z. (A set Λ ⊂ Z is syndetic if Z can becovered by a finite number of translations of Λ). Thanks to the invarianceof H(ϕ, ε) under translations, this characterization says that one can paveLϕ if and only if one can pave it by translations of a certain syndetic set.The general method in chapter 3 is to consider H(ϕ, ε) as a subset of Y =0, 1Z, which together with the translation (Tx)(n) = x(n + 1), becomesa dynamical topological system. It turns out that H(ϕ, ε) is a translation-invariant compact subset of Y (in the product topology), which thanks toresults from local Banach Space theory and dynamical topological systemtheory, namely, results by Bourgain-Tzafriri and Rusza, is rich enough tocontain sets of positive density.

The last chapter collects several equivalent formulations of the Kadison-Singer problem. We have added two new ones. Using an argument due toKeith Ball ([3]) we can show that the following formulation is equivalent tothe KS problem.

Restricted Invertibility. There exists a universal constant c > 0 and anatural number m such that for every linear T : ln2 → ln2 satisfying ||Tei|| = 1for each i = 1, 2, . . . , n, there exists a partition σ1, . . . , σm of 1, 2, . . . , nsuch that

||TPσix|| ≥ c||Pσix||

In chapter 4 we will show that restricted invertibility is equivalent to thepaving property:

Paving Property. Every operator T ∈ B(l2(N)) has the following property,called the paving property:

inf||n∑j=1

Pj(T − E(T ))Pj || :n∑j=1

Pj = Id, Pi ⊥ Pj ∀i 6= j = 0 (1.1)

All vector spaces in this work are assumed to be over the field of complexnumbers.

1.2 The Kadison-Singer Problem

Kadison and Singer were interested in the uniqueness of extension of ho-momorphisms defined on maximal abelian C∗ - sub-algebras of B(H) (thespace of bounded operators on a separable Hilbert space H), to positivefunctionals of the algebra B(H). A homomorphism of a C∗-algebra A is alinear and multiplicative function h, satisfying h(x) = h(x∗) for each x ∈ A.In particular, f(e) = 1, where e is the unit of A. A positive functional on

1.3. PURE STATES IN QUANTUM MECHANICS 7

a C∗-algebra A is a linear functional f satisfying f(e) 6= 0 and f(x∗x) > 0for each x ∈ A. The following problem was raised by Kadison and Singer in1959 [14].

Problem 1.2.1. Let A be a maximal abelian C∗ sub-algebra of B(H). Doesevery homomorphism of A have a unique extension to a positive functionalof B(H)?

As we shall see in the next chapter, the problem of uniqueness is themuch harder one; existence is guaranteed.

Kadison and Singer showed that in general the answer is negative. Theyproved that if A is isomorphic to L∞(0, 1) then there are homomorphismsof A that have more than one extension to a positive functional of B(H).They also conisdered the case where A is isomorphic to l∞(N) (or l∞(Z)),and in that case they did not settle the uniqueness problem.

The algebra l∞(N) is isomorphic to the diagonal algebra D ⊂ B(l2(N)),of all operators having the standard basis vectors as eigenvectors. The mem-bers of D are called diagonal operators. For an operator D ∈ D, the entriesdm,n = 〈Den, em〉 will vanish for n 6= m. The homomorphisms of D arewell known. For example, for each n ∈ N the functional ωn(D) = 〈Den, en〉is a homomorphism of D, and every homomorphism ρ on D is given byρ(D) = limU 〈Den, en〉, where U is an ultra-filter above N, determined byρ. We shall discuss the homomorphisms of D in more detail in the nextchapter.

One way to obtain extensions of homomorphisms of D, is to look at themap E : B(l2(N))→ D, mapping every operator T to the diagonal operatorE(T ) defined by: 〈E(T )en, em〉 = δnm〈Ten, em〉 where δnm = 1 if n = m, andzero otherwise. E is an order preserving projection of norm 1, and for everyhomomorphism ρ on D, ρE is a positive functional of B(l2(N)). Of course,the question is whether this is the only way to obtain homomorphisms of D.

1.3 Pure states in Quantum Mechanics

It is well known that the set of homomorphisms of an abelian C∗-algebra isprecisely the set of extreme points among the positive functionals on thatalgebra, normalized by f(e) = 1. That the normalized positive functionalsdo have extreme points follows from the fact that they form a convex andw∗-compact subset of the dual Banach-space of the C∗-algebra.

Normalized positive functionals on a C∗-algebra are called states, andthe extreme points among the states are called pure states. The choice ofthese names has to do with a physical interpretation of normalized positivefunctionals.

Physicists attach to certain physical systems in quantum mechanics aseparable Hilbert space H in such a way as to establish a one-to-one cor-respondence between observables of the system, (that is - those physical


quantities of the system that can be measured, like momentum and energy)and self-adjoint operators on H, and another one-to-one correspondence be-tween states of the system and bounded, positive operators onH whose traceis equal to 1. These latter operators are called density operators. The corre-spondence is such that given a Borel-measurable set ∆ ∈ R, the probabilityp(∆, T ) that a measurement of an observable corresponding to an operatorT when the system is in a state corresponding to a density-operator U , willbelong to the set ∆, is equal to

p(∆, T ) = tr(P T∆U) =∞∑n=1

〈P T∆Uen, en〉 (1.2)

where P T∆ is the spectral projection that corresponds to ∆, T , and en∞n=1

is an orthonormal basis in H. A state of the system for which the corre-sponding density-operator is a projection is called a pure state. If U is aprojection, then in order that it’s trace will be equal to 1 it needs to be aone-dimensional projection, so that U = x⊗ x for some unit vector x ∈ H.Recall that x ⊗ x is a one-dimensional projection on the span of x definedby: x ⊗ x(y) = 〈y, x〉x. In a pure state, the probability of equation (1.2)takes the form:

p(∆, T ) = tr(P T∆x⊗ x) = 〈P T∆x, x〉 (1.3)

In an attempt to clarify the notion of a pure state, let us observe thatgiven a unit vector x ∈ H for which equation (1.3) holds, we may constructa maximal abelian C∗ sub-algebra of operators all of which have x as aneigenvector. To do this, begin first with the algebra generated by x⊗ x andthe identity operator. This is an abelian C∗-algebra, and so it is contained insome maximal abelian C∗-algebra, in which every member commutes withx⊗ x, so every member has x as an eigenvector. Physicists say that observ-ables corresponding to commuting operators are compatible, meaning theycan be measured simultaneously. According to Dirac, [6], x is an eigen-vector of T if and only if a measurement of T in a state corresponding tox⊗ x, yields the value 〈Tx, x〉 with probability one. Consequently, there ismaximal information about a physical system when it is in a pure state.

Henceforth we shall only discuss bounded operators. In this case, if Uis a density-operator, then for each T ∈ B(H), UT has finite trace, so theequation ωU (T ) = tr(UT ) defines a norm-1 linear functional. Moreover, ω isa positive functional, and normalized, as ω(I) = tr(U) = 1. Such functionalsare called normal states. It can be shown ([7]) that the normal states form aconvex set whose extreme points are precisely the operators ωx⊗x, which wedenote for brevity by ωx. This fact motivates the definitions that we gaveprior to the physical discussion.

1.4. PURE STATES AND REPRESENTATION THEORY 9

1.4 Pure states and representation theory

Not all pure states of a C∗-algebra are ωx for some unit vector x ∈ H.Indeed, for the diagonal algebra D, we only have a countable sequence ofpure states of the form ωx, namely, ωen for the standard unit vectors, whileon the other hand we have uncountably many homomorphisms on D (thespace of homomorphisms of D is βN - Stone-Cech compactification of N;see the next chapter). Pure states not of the form ωx are called singularstates. A pure state is singular if and only if it annihilates all compactoperators. One direction is clear -if a pure state is not singular then it isωx for some unit vector and it of course does not vanish on the compactoperator x⊗ x. The proof of the other direction, that a singular pure statevanishes on all compact operators, requires some tools from representationtheory of C∗-algebras. See, e.g., [13], vol. II.

For an abelian C∗-algebra, the set of homomorphisms coincides with thepure states. As we shall presently see, every pure state can be extended to apure state of B(H). Therefore, an alternative formulation to Problem 1.2.1,which is also the original formulation of the problem, is the following:

Problem 1.4.1. Let A be a maximal abelian C∗ sub-algebra of B(H). Canevery pure state of A be uniquely extended to a pure state of B(H)?

The answer, as was already mentioned, is in general negative., but forA = D, H = l2(N), is still open. In this case the problem has implications onthe central problem of C∗-algebra theory, namely the problem of classifyingirreducible representations of B(l2(N)). A positive answer to the KS problemfor A = D will characterize, up to equivalence, a class of irreducible repre-sentations of B(l2(N), as follows: a theorem by Gelfand-Neymark states thatgiven a positive functional f on B(l2(N)), there is a cyclic representationT → πT of B(l2(N)) in a Hilbert space H

′such that

f(T ) = 〈πT ξ, ξ〉, ξ ∈ H′, ||ξ|| = 1, T ∈ B(l2(N)) (1.4)

If in (1.3) f is a state of B(l2(N)) then the corresponding representation isirreducible if and only if f is a pure state. ([13], Vol. II, Th. 10.23). Letus examine under what conditions the restriction of a pure state f to thediagonal algebra D is a pure state of D. If the restriction fD is a pure state,then on D f is multiplicative so for all D ∈ D:

|f(D)| = |〈πDξ, ξ〉| ≤ ||πDξ||= 〈πDξ, πDξ〉1/2 = 〈πD∗Dξ, ξ〉1/2

= f(D∗D)1/2 = (|f(D)|2)1/2 = |f(D)|

Since we have equality in Schwarz’ inequality, we infer that ξ is an eigenvec-tor of all operators πD, D ∈ D, with corresponding eigenvalue f(D). In other


words, the restriction of the representation π to D gives a one-dimensionalrepresentation of D in the space spanned by ξ. On the other hand it isclear that if ξ is an eigenvector of all the operators πD then the functionalf(D) = 〈πDξ, ξ〉 is multiplicative, hence pure, as a state of D. (Along theselines we can show that if A is an abelian C∗-algebra of operators on a Hilbertspace H, then the state ωx = 〈Tx, x〉 is a pure state of A if and only if x isan eigenvector of all the operators in A. ) We have proved that wheneverf is a state of B(l2(N)), the restriction of f to D is a pure state if and onlyif the restriction of the representation given by (1.4) is one-dimensional.Subsequently, an affirmative answer to the KS problem will yield that therepresentation in (1.4) is irreducible if and only if it’s restriction to D isone-dimensional. These considerations show that the problem of classifica-tion of irreducible representations of B(l2(N)) is more general than the KSproblem. A solution of the former will solve the latter, but even if the KSproblem were to be solved affirmatively, there are not enough pure states ofD in order to characterize all the irreducible representations of B(l2(N)) byconsidering their restrictions to D. Indeed, not all pure states of B(l2(N))are extensions of pure states of D. To see this, it suffices to take a state ofthe form ωx where x is a unit vector which is not a standard basis vector.x is not an eigenvector of all the diagonal operators, and so the restrictionof ωx to D cannot be a pure state. Nevertheless, ωx is a pure state of thebigger algebra B(l2(N)). A short way to see this is to first prove:

Theorem 1.4.1. Let A be a maximal abelian C∗ sub-algebra of B(H). Ifthe state ωx(T ) = 〈Tx, x〉 is a pure state of A, then it has a unique extensionto a pure state of B(H).

Proof. This proof is taken from [14]. The projection x ⊗ x commutes withevery operator for which x is an eigenvector. Therefore x ⊗ x commuteswith all operators of A, and since A is maximal, x⊗ x must belong to A. Ifω is any extension of ωx to a state of B(H), then ω is not singular becauseit does not vanish on the compact x ⊗ x. Therefore there is a unit vectory ∈ H such that ω = ωy. On A ωy = ωx so

1 = ωx(x⊗ x) = ωy(x⊗ x) = 〈〈y, x〉x, y〉 = |〈x, y〉|2 (1.5)

So y belongs to the range of x⊗x, which means that y = tx for some t, andsince ||x|| = ||y|| = 1, we must have |t| = 1. As a result, for every T ∈ B(H):

ωy(T ) = 〈Ty, y〉 = |t|2〈Tx, x〉 = ωx(T ) (1.6)

which shows that ωx is the unique extension of its restriction to A.

Armed with this result, in order to show that ωx is a pure state ofB(l2(N)) we construct, as before, a maximal abelian C∗ sub-algebra A of

1.4. PURE STATES AND REPRESENTATION THEORY 11

operators all having x as an eigenvector. The restriction of ωx to A is a purestate, and so by Theorem 1.4.1 ωx itself is a pure state.

Every pure state of A has at least one extension to a pure state ofB(l2(N)). Theorem 1.1 shows that the states ωen on D have a unique exten-sion to pure states of B(l2(N)). We shall see in the next chapter how thisresult can be obtained in a different way.

Chapter 2

The pure states of D

2.1 Preliminaries

A C∗-algebra is a Banach-algebra A over the complex field C, equipped witha unit e and an involution operation x→ x∗ such that ||x∗x|| = ||x||2. Thereare two main types of C∗ algebras: the abelian ones, modeled as an algebraC(K) of continuous complex-valued functions on some compact Hausdorfspace K, with the maximum norm and an involution operation given byconjugation: f → f , and the non-abelian ones, modeled as the algebraB(H) of bounded linear operators on some Hilbert space H, with the usualnorm and involution furnished by the adjoint operation T → T ∗. EveryC∗ algebra A is isomorphic to a closed sub-algebra of B(H) for a suitableHilbert space H, and the isomorphism preserves involution, that is, if x ∈ Ais mapped to φx, then x∗ is mapped to (φx)∗. ([15], Th. 12.41). AbelianC∗ algebras are ∗-isomorphic and isometric to C(∆), where ∆ is the spaceof homomorphisms of A. A homomorphism of A is a linear multipilicativefunctional on A, and the topology on ∆ is that induced by the w∗-topologyof A

′, the dual space of A.

A linear functional f on a C∗ algebra A is called positive if f(x∗x) ≥ 0for every x ∈ A. A convenient criterion for checking positivity of a linearfunctional f is this: f is positive if and only if f is bounded and ||f || = f(e),([13] Th. 4.3.2.). For example, if H is a Hilbert space and y ∈ H thenf(T ) = 〈Ty, y〉 is a positive functional on B(H). If ||y|| = 1 then ||f || = 1.Denote by S(A) the set of positive linear functionals on a C∗ algebra A,satisfying ||f || = f(e) = 1. These are the states of A.

Proposition 2.1.1. Let A be a C∗ algebra. The set of normalized positivefunctionals S(A) is convex and w∗ compact in A

′. Let P (A) denote the set

of extreme points of S(A). Then A is abelian if and only if P (A) consistsprecisely of the homomorphisms of A.

Proof. By the Hahn-Banach theorem, there exists a bounded linear func-tional f such that f(e) = ||e|| = 1, so S(A) 6= ∅. If f, g ∈ S(A) and 0 < t < 1

13

14 CHAPTER 2. THE PURE STATES OF D

then for each x ∈ A

(tf + (1− t)g)(x∗x) = tf(x∗x) + (1− t)g(x∗x) ≥ 0

and (tf + (1 − t)g)(e) = 1, so S(A) is convex. If F ∈ S(A)w∗

, then givenε > 0 the set

Wx = G ∈ A′ : |G(e)− F (e)| < ε ∩ G ∈ A′ : |G(x∗x)− F (x∗x)| < ε

is w∗-open, containing F , and intersecting S(A) for every choice of x ∈ A.Thus there exist f, g ∈ S(A) such that |f(e) − F (A)| < ε and |g(x∗x) −F (x∗x)| < ε so F (x∗x) ≥ 0 for all x ∈ A and F (e) = 1. Hence, S(A)

w∗= S(A).

S(A) is norm-bounded and w∗-closed, so by the Banach-Alauglo the-orem, it is w∗-compact. By the Krein-Milman theorem, P (A) 6= ∅. IfA is abelian, A is ∗-isomorphic and isometric to C(∆) where ∆ is thehomomorphism-space of A. The equation

F (x) =

∫∆x dµ (2.1)

defines a 1-1 correspondence between S(A) onto the set of probability mea-sures on ∆, where x is the Gelfand transform of x ∈ A. ([15], Th. 11.32).Extreme points of S(A) correspond to measures concentrated at points of∆ and vice versa. Thus P (∆) = ∆. If P (∆) is the homomorphism spaceof A, then for each x, y ∈ A and f ∈ P (A), f(xy − yx) = 0. This willimply that xy = yx. Without loss of generality, A is a closed sub-algebra of

B(H) for a suitable Hilbert space H. Since S(A) = convP (A)w∗

, we haveF (xy − yx) = 0 for each F ∈ S(A). Let ξ ∈ H be a unit vector. Then〈(xy − yx)ξ, ξ〉 = 0 and so 〈xyξ, ξ〉 = 〈yxξ, ξ〉 for all unit vectors ξ ∈ H.Hence xy = yx, and A is abelian.

Remark 2.1.1. In general P (A) is not w∗-closed in A′. Indeed, if H is an

infinite dimensional Hilbert space, then P (B(H)) is not w∗-closed; see [13],vol. I 4.6.69.

2.2 The pure states of the algebra l∞(N)

Take A = l∞(N) over C. P (A) can be identified with βN, the Stone-Cechcompactification of N. To see this, take f ∈ P (A) and x ∈ A. Then f(x) liesin the spectrum of x. (Proof: If not, then x− f(x)e is invertible with someinverse y, and then 1 = f(e) = f((x − f(x)e)y) = f(x − f(x)e)f(y) = 0).Now, the spectrum of x is x(n)∞n=1 and so given ε > 0 and x1, . . . , xk ∈ A,there is an n ∈ N such that

max1≤j≤k

|f(xj)− xj(n)| < ε (2.2)

2.2. THE PURE STATES OF THE ALGEBRA L∞(N) 15

Denote by ωn the homomorphism of A defined by ωn(x) = x(n). (2.2)shows that for any f ∈ P (A), every w∗-neighborhood of f meets ωnn∈Nand so P (A) ⊂ ωn

w∗

n∈N. On the other hand, P (A) is w∗-compact, and

ωn ∈ P (A) for all n ∈ N, hence P (A) ⊃ ωnw∗

n∈N, so we have inclusions in

both directions and P (A) = ωnw∗

n∈N. We shall make use of the identificationbetween the pure states on A and the collection of ultra-filters in N.

Theorem 2.2.1. Every ultra-filter U in N defines a homomorphism f ∈P (A) by fU (x) = limU x(n), and given a homomorphism f ∈ P (A) thereexists a unique ultra-filter U in N such that f = fU .

Proof. To construct U , define for each subset σ ⊂ N a member xσ of l∞ by

xσ = 1⇐⇒ n ∈ σ (2.3)

and define a family U of subsets by:

U = σ ⊂ N : f(xσ) = 1 (2.4)

Since for each σ: x2σ = xσ, the fact that fσ is multiplicative implies f(xσ) =

f(x2σ) = f(xσ)2, and so either f(xσ) = 0 or f(xσ) = 1. U is the desired

ultfa-filter: if σ ∈ U , and τ some set containing σ, then xσ ≤ xτ , and sof(xσ) ≤ f(xτ ), and f(xτ ) = 1. If σ, τ ∈ U then f(xσxτ ) = f(xσ)f(xτ ) = 1and xτ · xσ = xτ∩σ, so that the intersection τ ∩ σ belongs to U . Clearly theempty set is not a member of U . These considerations already show that Uis a filter, and in order to show that it is actually an ultra-filter, it sufficesto show that for each σ ⊂ N, exactly one of the sets σ, σc is contained inU , and this follows immediately from the equality f(xσ + xσc) = f(e) = 1.It remains to show that limU x(n) = f(x) for all x ∈ A. (The limit existsbecause x(n))∞n=1 is bounded). Recall that for an ultra-filter U the limitlimU x(n) is equal to some y if and only if for every open set O containingy, the set n : x(n) ∈ O belongs to U . So we need to show that for eachε > 0 the set σ = n ∈ N : |x(n) − f(n)| < ε belongs to U , that is,f(xσ) = 1. σ is not empty, because as was remarked earlier f(x) belongs tothe spectrum of x which is the closure of the sequence x(n)∞n=1. Let xσdenote the Gelfand transform of xσ. Since x2

σ = xσ and xσ is continuous,xσ must be the characteristic function in P (A) of the closed and open setx−1σ 1. If xσ(f) = 0, then there is a neigbhorhood W ⊂ P (A) of f such

that for each g ∈ W , xσ(g) = 0. But the topology of P (A) is homemorphicto the w∗-topology of the dual A

′, and since f(x) ∈ x(n)∞n=1, we deduce

that every w∗-open set containing f contains also a homomorphism of theform ωn, where n may also be picked from the set σ. As a result, sincexσ(ωn) = 1, we cannot have xσ(f) = 0, and so xσ(f) = f(xσ) = 1.


2.3 Extensions of pure states of D

We mentioned in the introduction that the statement ”all operators in B(H)have the paving property” is equivalent to the statement ”all pure states of Dhave a unique extension to a pure state of B(H)”. It was shown by Kadisonand Singer that any two extensions ρ1, ρ2 of a pure state of D to positivefunctionals of B(H) coincide on all operators that have the paving property.It is natural to seek a characterization of those pure states of D that dohave a unique extension to pure states of B(H). This is the purpose of thissection. The main argument is taken from [14]. Anderson also proved aresult along these lines. ([1]). We shall need a proposition and a lemma.

Proposition 2.3.1. Let A,B be C∗-algebras, B ⊂ A, and e ∈ B.

1. Every state f ∈ S(B) has an extension to a state f ∈ S(A).

2. Every pure state f ∈ P (B) has an extension to a pure state F ∈ S(A).

3. A pure state f ∈ P (B) has a unique extension to a pure state F ∈P (A) if and only if f has a unique extension to a state F ∈ S(A).

4. A pure state f ∈ P (B) has a unique extension to a pure state F ∈P (A) if and only if for all y ∈ A, y = y∗:

supf(x) : x = x∗ ∈ B, x ≤ y = inff(x) : x = x∗ ∈ B, x ≥ y

The proofs of (1)-(3) are simple, whereas (4) is the main tool to obtainnecessary and sufficient conditions for the uniqueness of extension of a par-ticular pure state. This is in fact part of Theorem 4.3.13 in [13], and theproof can be found there.

Proof. (1) The Hahn-Banach theorem gives an extension F of f ∈ S(B)so that ||F || = ||f ||. As F (e) = 1 = ||f || = f(e), F is a norm-1 positivefunctional, that is, a state.

(2) Let f ∈ P (B), and denote by E(f) the set of extensions of f that donot increase it’s norm. (1) Shows that E(f) is not empty; E(f) is convex,and we skip the routine check. Moreover, E(f) is w∗-closed: If F ∈ E(f)w∗

then for each x ∈ B and ε > 0 we can find G ∈ E(f) such that |G(x) −F (x)| < ε, and so |f(x)−F (x)| < ε. Since ε is an arbitrary positive number,we get f(x) = F (x), and this holds for all x ∈ B. Consequently, F extendsf , and E(f) is therefore convex and w∗-compact, so it has extreme points.We will show that all it’s extreme points are in fact pure states of A, and thiswill conclude the proof of (2). Assume F is an extreme point of E(f) butnot a pure state of A. Then there are states g, h ∈ S(A) so that F = g+h

2 .

Restrict this equality to B: f|B =g|B+h|B

2 . Since f is a pure state on B, itdoes not have non-trivial convex representations there. So g|B = h|B = f ,

2.3. EXTENSIONS OF PURE STATES OF D 17

and so g, h are also extensions of f , that is, g, h ∈ E(f). However, we didassume g 6= h, which contradicts F being an extreme point of E(f). So allextreme points of E(f) are pure states of A.

(3) If f ∈ P (B) has a unique extension, then E(f) has only one extremepoint, and by the Krein-Milman theorem so does E(f). The other directionis also immediate.

Lemma 2.3.1. Let A,B be C∗-algebras, B ⊂ A, and B abelian. If F ∈S(A) is an extension of a homomorphism f ∈ P (B), then for every y ∈A, x ∈ B:

F (xy) = F (yx) = F (y)f(x)

Proof. There exists a cyclic representation πF of A in some Hilbert space H,such that for every y ∈ A, πF (y) belongs to H, and the connection betweenf and πf is given by:

F (y) = 〈πF (y), ξ〉 ξ ∈ H, ||ξ|| = 1 (2.5)

where ξ ∈ H is a cyclic vector for the representation y → πF (y). For everyx ∈ B:

|f(x)| = |〈πF (x)ξ, ξ〉| ≤ ||πF (x)||||ξ|| = 〈πF (x)ξ, πF (x)ξ〉1/2

= 〈πF (x∗x)ξ, ξ〉1/2 = f(x∗x)1/2 = |f(x)|

Thus we have equality in Schwartz inquality, so there must be some scalarλ ∈ C for which πF (x)ξ = λξ. Comparing with (2.5) we see that λ = F (x) =f(x). Therefore for every x ∈ B, y ∈ A:

F (xy) = 〈πF (xy)ξ, ξ〉 = 〈πF (x)πF (y)ξ, ξ〉= f(x)〈πF (y), ξ〉 = f(x)F (y)

A similar calculation for F (yx) yields the same result.

For a subset σ ⊂ N, let Pσ denote the diagonal projection defined by:

n ∈ σ ⇐⇒ 〈Pσen, en〉 = 1

Theorem 2.3.1. Let f ∈ P (D). A necessary and sufficient condition that fhas a unique extension to a state F ∈ S(B(l2)) is that for every T ∈ B(l2):

inf ||Pσ(T − E(T ))Pσ|| : σ ∈ Uf = 0

where Uf is the ultra-filter corresponding to f , and E(T ) the diagonal of T .


Proof. The condition is sufficient: In the notation of proposition 2.3.1, letf ∈ E(f). By the lemma, for every σ ∈ Uf and every T ∈ B(l2):

βf (T ) = F (Pσ(T − E(T ))Pσ) = (f(Pσ))2F (T − E(T ))

= f(P 2σ )F (T − E(T ))

= F (T − E(T ))

so

|F (T − E(T ))| = |F (Pσ(T − E(T ))Pσ)| ≤ ||F ||||Pσ(T − E(T ))Pσ||= ||Pσ(T − E(T ))Pσ||

Therefore if βf (T ) = 0 then F (T ) = F (E(T )) = fE(T ), so that if for everyT ∈ B(l2), βf (T ) = 0 then F = fE is the only extension of f .

The condition is necessary: From the assumption it follows that F = fEis the only extension of f . Let T ∈ B(l2) and assume at first that T is self-adjoint and it’s diagonal is zero. From proposition 2.3.1(4), we have:

0 = supf(D) : D = D∗, D ≤ T = inff(D) : D = D∗, D ≥ T

and so given ε > 0, we can find self-adjointD1, D2 ∈ D, so thatD1 ≤ T ≤ D2

and

−ε < f(D1) ≤ 0 ≤ f(D2) < ε (2.6)

and from the continuity of D1, D2 on P (D) we deduce that there existsa neighborhood W of f in P (D) so that for each g ∈ W inequality (2.6)holds with g in place of f . Since P (D) is the Stone-Cech compactificationof the natural numbers, P (D) is totally disconnected, whence the closure ofeach open set is open. Therefore the boundary of W is not empty and so theindicator function χW is continuous. Let PW denote the diagonal projection

satisfying PW = χW . Put:

σ = n ∈ N : 〈PW en, en〉 = 1

then PW = PW . Moreover, if Uf is the ultra-filter correpsonding to f , thenσ ∈ Uf , because

f(Pσ) = f(PW ) = Pσ(f) = PW (f) = χW (f) = 1

Now,

εPσ ≥ PσD2 = PσD2Pσ ≥ PσTPσ ≥ PσD1Pσ = PσD1 ≥ −εPσ

2.4. GLIMM’S CONJECTURE 19

since for self-adjoint operators S, T the inequality −T ≤ S ≤ T implies||S|| ≤ ||T || ([13], 4.2.8), we have:

||PσTPσ|| ≤ ε||Pσ|| = ε (2.7)

Since for each ε > 0 we can find σ ∈ Uf such that (2.7) holds, we musthave βf (T ) = 0. If T is not self-adjoint, we can write T = T1 + T2 withself adjoint T1, T2, and T −E(T ) is an operator whose diagonal is zero. Thegeneral case follows easily.

The preceding theorem facilitates a simple proof of the next result.

Corrolary 2.3.1. For each n ∈ N, the pure state defined on D by ωn(D) =〈Den, en〉 has a unique extension to a pure state of B(l2), given by Ωn(T ) =ωn(E(T )).

Proof. The corresponding ultra-filter to ωn is the familty of subsets σ ⊂ Ncontaining n. Since n ∈ Uωn , for each T ∈ B(l2) we have

βωn(T ) ≤ ||Pn(T − E(T ))Pn|| = ||〈Ten, en〉I − 〈E(T )en, en〉I|| = 0

where I denotes the identity operator, and so by the preceding theorem, ωnhas a unique extension.

At this point arises the question whether there exist pure states on D thatcorrespond to points in βN−N and have a unique extension to a pure stateof B(l2). Reid answered this question in the affirmative ([15]). He showedthat if U is an ultra-filter having the following property: For every partitionof N into disjoint intervals [1, n1], [n1 +1, n2], . . ., there exists a subset σ ∈ Uso that σ contains precisely one member of each partition-interval, then thepure state that corresponds to this ultra-filter has a unique extension to apure state of B(l2). At this time we know of no other examples of ultra-filters corresponding to points of βN− N for which the corresponding purestate has a unique extension to a pure state of B(l2).

2.4 Glimm’s conjecture

Definition 2.4.1. Let A,B be C∗-algebras, A ⊂ B. We say that A sep-arates the pure states of B if given f, g ∈ P (B), f 6= g, there exists somea ∈ A such that f(a) 6= g(a).

We shall not enter here into a detalied discussion of C∗- algebras separat-ing pure states of C∗-algebras that contain them. This topic is investigatedin the general theory of C∗-algebras, and not all problems are solved. Forexample, in [9], Glimm conjectures:


Conjecture 2.4.1. Let A,B be C∗-algebras, A ⊂ B. If A separates thepure states of B, then B = A.

Separation of pure states is clearly connected to the KS problem: IfA,B are C∗-algebras, A ⊂ B and A separates the pure states of B, thenevery pure state of A has a unique extension to a pure state of B. Here weshow:

Proposition 2.4.1. An abelian C∗ sub-algebra never separates the purestates of a larger C∗-algebra. More precisely: Let A,B be C∗-algebras, A ⊂B and B 6= A. If A is abelian, then A does not separate the pure statesof B.

For the proof we need two lemmas.

Lemma 2.4.1. If A,B are C∗-algebras, A ⊂ B and A separates the purestates of B, then the restriction of each pure state of B to A is a pure stateof A.

Proof. We use the following characterization of pure states: A state f of aC∗-algebra is pure if and only if the representation πf corresponding to f bythe Gelfand-Neymark-Segal theorem (the GNS representation) is irreducible.(πf is given by f(x) = 〈πf (x)ξ, ξ〉 where ξ is a normalized vector in therepresentation space.) A proof of this characterization appears, e.g., in [13],II, 10.23.

Let f ∈ P (B). There exists a Hilbert space Hf and a cyclic repre-sentation x → πf (x) so that f(x) = 〈πf (x)ξ, ξ〉 where ξ ∈ Hf is a cyclicunit vector. Citing the characterization above, x → πf (x) is an irreduciblerepresentation. The restriction of f to A is a representation of A given by:

f(x) = 〈πf (x)ξ, ξ〉 (x ∈ A) (2.8)

To see that the restriction f|A is a pure state of A, it suffices to show thatπf in (2.8) is irreducible. Let H ⊂ Hf be a closed subspace of Hf , invariantunder the action of the operators πf (x) : x ∈ A. Let E be the orthogonalprojection on H. Take y ∈ H, z ∈ H⊥ with ||y + z|| = 1. Then for eachx ∈ A

〈πf (x)(y + z), (y + z)〉 = 〈πf (x)y, y〉+ 〈πf (x)z, z〉 (2.9)

From this we see that πf (x)y ∈ H for every x ∈ A and H⊥ is also invariantunder the operators πf (x) : x ∈ A. Moreover, we have

∀x ∈ A 〈πf (x)(y − z), (y − z)〉 = 〈πf (x)y, y〉+ 〈πf (x)z, z〉 (2.10)

For x ∈ B, define f1, f2 by

f1(x) = 〈πf (x)(y + z), (y + z)〉 (2.11)

2.4. GLIMM’S CONJECTURE 21

f2(x) = 〈πf (x)(y − z), (y − z)〉 (2.12)

Every non-zero vector in Hf is a cyclic vector for the representation x →πf (x), and y ⊥ z, y±z 6= 0, so (2.11) and (2.12) define irreducible representa-tions of B, so f1, f2 are pure states of B. As their restrictions to A coincide,the assumption that A separates points in B implies that f1 = f2. A re-sult from representation theory ensures there is a λ ∈ C, |λ| = 1 such thaty+z = λ(y−z). Hence, either y = 0 or z = 0. Since y ∈ H, z ∈ H⊥ were ar-bitrary, we infer that either H = 0 or H = Hf , showing that Hf does nothave non-trivial subspaces invariant under the operators πf (x) : x ∈ A.The representation (2.8) is therefore irreducible.

As was remarked in the introduction, there exist pure states of B(l2)whose restrictions to D are not pure states. For example, if x = e1+e2√

2with

e1, e2 the first two standard unit vectors, then x is a unit vector which isnot an eigenvector of the diagonal operator Dx = 〈x, e1〉 and so ωx|D is nota pure state of D. This phenomenon is not special to D ⊂ B(l2).

Lemma 2.4.2. Let A be a maximal abelian C∗ sub-algebra of a C∗-algebraB, and A 6= B. There exists a pure state of B whose restriction to A is notpure.

Proof. If not, then for every pure state f ∈ P (B) the restriction f|A is apure state of A. From lemma 2.3.1 we deduce that for each x ∈ B, y ∈ Aand every pure state f ∈ P (B), f(xy) = f(yx), so yx = xy, and from themaximality of A, A = B, contrary to the assumption.

Proof of proposition 2.4.1

If B is not abelian, then by the preceding lemmas we may replace A aby maximal abelian algebra A

′such that A ⊂ A

′ ⊂ B, and since A′

doesnot separate points in P (B), neither does A. If on the other hand B isabelian, then since B can be identified with C(P (B)) and A with C(P (A)),the fact that A 6= B implies that the inclusion C(P (A)) ⊂ C(P (B)) is alsostrict. By Stone-Weirstrass, C(P (A)) does not separate the points of P (B).Consequently, there exist f1, f2 ∈ P (B) with f1 6= f2 so that f1(g) = f2(g)for each g ∈ C(P (A)), where g is the Gelfand transform of g ∈ P (A). As aresult, f1(g) = f2(g) for every g ∈ P (A), and A does not separate the purestates of B.

Example 2.4.1. Proposition 2.4.1 shows that whenever an abelian C∗ algebrais injected into a larger C∗-algebra B, then the algebraic property of beingabelian implies that A is not large enough to separate all the pure states ofB. As in other cases in the theory of C∗-algebras, the algebraic property hasa topological consequence. To illustrate this, take B = B(l2) and A = D.Let E : B(l2) → D denote the projection carrying an operator T to it’sdiagonal E(T ). For every f ∈ P (B), fE is a state of B(l2). This fact


follows from properties of the projection E: E is order preserving in thesense that if T ≤ S then E(T ) ≤ E(S) and ||E|| = 1. Consider the mapΛ : P (D) → B(l2) given by Λf = fE. Λ maps the compact space P (D)onto a compact subspace (in the w∗-topology) of S(B(l2)) - the state spaceof B(l2), where the topology of P (D) is the Gelfand topology, i.e., the w∗

topology of P (D) as a subset of D′ - the dual space of D. To see this, takef ∈ P (D) and a w∗-open set U containing Λf , say, of the form

U = F ∈ (B(l2))′

: |F (Ti)− Λf(Ti)| < ε i = 1, 2, . . . , n

where Ti are in B(l2). If we take

V = h ∈ P (D)|h(D(Ti))− Λf(Ti)| < ε i = 1, 2, . . . , n

then V is an open set containing f ∈ P (D) and if h ∈ V then

|Λh(Ti)− Λf(Ti)| = |h[E(Ti)]− Λf(Ti)|

whence ΛV ⊂ U , so Λ is continuous. If Λ(f1) = Λ(f2) for f1, f2 ∈ P (D)then for every T ∈ B(l2) f1(E(T )) = f2(E(T )) so f1, f2 coincide on D,whence f1 = f2. This shows that Λ : P (D) → S(B(l2)) is a continuousinjection of the pure states of D into the states of B(l2). Now, the set ofstates of a separable Hilbert space is never w∗-closed ([13]), and certainly isnot w∗-compact. Therefore, even if the KS conjecture holds, and every purestate of D does indeed have a unique extension to a pure state of B(l2), inwhich case the unique extension is explicitly given by a composition withthe diagonal, fE, even in that case there will always be pure states of B(l2)that cannot be obtained as extensions of pure states of D - and this is dueto topological reasons: P (B(l2)) is not w∗ compact, but P (D) as well as it’sembedding into S(B(l2)) are indeed compact.

To end this chapter, let us remark that there is always an order-preservingprojection Θ : B → A when A is maximal abelian in a C∗-algebra B. Thiswas proved by Von-Neumann. Therefore we can always embed P (A) in-side S(B), as we did for D ⊂ B(l2), so that the topological phenomenondescribed above is general.

Chapter 3

The Laurent Operator

3.1 Basics

For each ϕ ∈ L∞(0, 2π), Lϕ is an operator from L2(0.2π) to itself, definedfor each f by Lϕf = fϕ. The matrix representing Lϕ with respect to thecharacters en(t) = eint, 0 ≤ t ≤ 2π is given by:

Lϕ(m,n) = 〈Lϕen, em〉 =1

2π

∫ 2π

0ϕ(t)e−i(m−n)t dt = ϕ(m− n) (3.1)

It is well known that the map ϕ → Lϕ is a ∗-isometric and isomorphicembedding of the C∗ algebra L∞(0, 2π) into B(L2(0, 2π)), so that the normof the Laurent operator Lϕ is just ||ϕ||L∞ . In fact, Lϕ is a normal operator,and so it’s norm equals it’s spectral radius ([15] Ex.20, p. 325), which isjust ||ϕ||L∞ . The operator-algebra L = Lϕ : ϕ ∈ L∞(0, 2π) is a maximalabelian C∗ sub-algebra of B(L2(0, 2π)). (Maximality is not trivial. For aproof, see [11] problem 115).

Kadison and Singer showed in [14] that there are pure states of L that donot have a unique pure state extension to B(L2(0, 2π)). In this context, Lis the algebra whose pure states are investigated. On the other hand, whenwe investigate the paving property of Laurent operators, it is rather thediagonal sub-algebra of B(l2(Z)) whose pure states we are looking at. Everytime we are looking at the paving property of a single Laurent operator, oran algebra of such operators, we should keep in mind that we are talkingabout uniqueness of extension of pure states of D, the diagonal sub-algebraof B(l2(N)) (or B(l2(Z))). We note in passing that whereas B(l2(Z)) is thenatural habitat of the Laruent operators’ matrix representations, the spaceB(l2(N)) naturally accommodates the Toeplitz operators Tϕ, obtained fromthe Laurent operators by applying the orthogonal projection onto the spaceof functions whose Fourier coefficients that correspond to negative integersall vanish.

23

24 CHAPTER 3. THE LAURENT OPERATOR

In the first chapter we introduced, for each T ∈ B(l2(N)), the number

βf (T ) = inf||Pσ(T − E(T ))Pσ|| σ ∈ Uf

where f ∈ P (D), and Uf is the corresponding ultra-filter. We may replaceN by Z, and then Uf will be an ultra-filter over Z. The next propositionfollows immediately from the definitions:

Proposition 3.1.1. For every S, T ∈ B(l2(Z)), f ∈ P (D), λ ∈ C and asequence Tn ∈ B(l2(Z)) such that Tn → T in the norm-topology, one has

1. βf (T + S) ≤ βf (T ) + βf (S)

2. βf (λT ) = |λ|βf (T )

3. limn→∞ βf (Tn) = βf (T )

Together with theorem 2.3.1 this proposition shows that the collectionof operators T ∈ B(l2(Z)) that have the paving property, forms a Banachspace. This comment justifies the fact that in the course of this chapter weshall restrict our attention to Laurent operators Lϕ whose ϕ is a real valuedfunction.

3.2 The family H(ϕ, ε)

Let Y = 0, 1Z. Here is a metric that makes Y compact:

d(x, y) = inf 1

k + 1: x(n) = y(n) |n| ≤ k (3.2)

Indeed, this metric induces the Tychonoff topology on Y . The longer theinterval on which x, y agree, the shorter is the distance between them. LetT denote the translation operator Tx(n) = x(n + 1). T is continuous withrespect to our metric, because if x, y agree on [−N,N ], then Tx, Ty agreeat least on [−N + 1,−N − 1].

We defined in the introduction the following family of subsets of Z:

H(ϕ, ε) = Λ ⊂ Z : ||PΛ(Lϕ − E(Lϕ))PΛ|| ≤ ε (3.3)

Our first task is to identify H(ϕ, ε) with a compact subset of Y , where weidentify subsets Λ ⊂ Z with x ∈ Y such that x(n) = 1 if and only if n ∈ Λ.We start with a simple lemma.

Lemma 3.2.1. If xn is a sequence in a Hilbert space H converging in normto y, then for every bounded operator T ∈ B(H), 〈Txn, xn〉 → 〈Ty, y〉 asn→∞.

3.2. THE FAMILY H(ϕ, ε) 25

Proof.

|〈Txn, xn〉 − 〈Ty, y〉| = |〈T (xn − y), xn − y〉+ 〈Txn, y〉+ 〈Ty, xn〉 − 2〈Ty, y〉|≤ |〈T (xn − y), xn − y〉|+ |〈T (xn − y), y〉|+ |〈Ty, xn − y〉|≤ ||T ||||xn − y||2 + ||T ||||xn − y||||y||+ ||T ||||y||||xn − y||

Lemma 3.2.2. Let ϕ ∈ L∞(0, 2π) real valued, and ε > 0. Then:

1. For Λ ⊂ Z, k ∈ Z:

||PΛ(Lϕ − E(Lϕ))PΛ|| = ||PΛ+k(Lϕ − E(Lϕ))PΛ+k||

2. Given natural N , put ΛN = Λ ∩ [−N,N ]. Then

||PΛ(Lϕ − E(Lϕ))PΛ|| = supN||PΛN

(Lϕ − E(Lϕ))PΛN||

= limN→∞

||PΛN(Lϕ − E(Lϕ))PΛN

||

3. Λ ∈ H(ϕ, ε) if and only if for every finite subset σ ⊂ Λ, σ ∈ H(ϕ, ε).

Proof. As ϕ is real valued, PΛ(Lϕ − E(Lϕ))PΛ, which is equal to PΛ(Lϕ −ϕ(0)I)PΛ is self-adjoint for every Λ ⊂ Z, so:

||PΛ(Lϕ − E(Lϕ))PΛ|| = sup||f ||≤1

|〈PΛ(Lϕ − Lϕ(0)I)PΛf, f〉| (3.4)

= sup||f ||≤1,f∈ImPΛ

|〈f · ϕ, f〉 − ϕ(0)||f ||2| (3.5)

since for every k, n ∈ Z and f ∈ L2(0, 2π), f e−k(n) = f(n + k), and sincewe also have the equalities:

(f ∈ L2(0, 2π), k ∈ Z) : 〈(fe−k)ϕ, fe−k〉 = 〈fϕ, f〉 (3.6)

and

(f ∈ L2(0, 2π), k ∈ Z) : ||fe−k|| = ||f || (3.7)

we deduce that 1. holds.

To prove 2, we note that for N ∈ N and Λ ⊂ Z, we have ΛN ⊂ ΛN+1,hence:

βN = ||PΛN(Lϕ − ϕ(0)I)PΛN

|| ≤ ||PΛN+1(Lϕ − ϕ(0)I)PΛN+1

|| = βN+1 (3.8)

so βN∞N=1 is a non-decreasing sequence of non-negative real numbersbounded by α = ||PΛ(Lϕ − ϕ(0)I)PΛ||. The proof of 2. will be complete


once we show that α = supN βN . Indeed, given δ > 0, (3.5) shows thatthere is some f0 ∈ ImPΛ, ||f0|| ≤ 1 such that∣∣|〈f0ϕ, f0〉 − ϕ(0)||f ||2| − α

∣∣ < δ/2

since SNf0 =∑

n∈ΛNf0(n)en is a sequence converging in L2(0, 2π) to f0,

the previous lemma implies that there is N > 0 for which∣∣|〈(SNf0)ϕ, SNf0〉 − ϕ(0)||SNf ||2| − α∣∣ < δ/2 (3.9)

from which it follows that

α− δ < |〈(SNf0)ϕ, SNf0〉 − ϕ(0)||SNf ||2|

and since SNf0 ∈ ImPΛ this implies:

α− δ < sup|〈gϕ, g〉 − ϕ(0)||g||2| ||g|| ≤ 1 g ∈ ImPΛN (3.10)

and the r.h.s of this last inequality is just ||PΛN(Lϕ − ϕ(0)I)PΛN

||. Thus αis indeed the supremum of βN∞N=1, and since βN is non-decreasing α isalso the limit of βN as N → ∞. The proof of 2. is complete. 3. followsimmediately from 2.: every finite subset of Λ is ΛN for all sufficiently largeN .

Now for each natural number N , consider the set

XN = x ∈ Y : n : x(n) = 1, |n| ≤ N ∈ H(ϕ, ε)

XN represents all subsets Λ ⊂ Z whose intersection with [−N,N ] belongsto H(ϕ, ε). Clearly, XN ⊃ XN+1. Moreover, XN is compact in the metricd(x, y), because if y ∈ XN , then there is an x ∈ XN such that d(x, y) < 1

N+1 .In particular, x(n) = y(n) for all |n| ≤ N so the set n : x(n) = y(n) alsobelongs to XN . Since XN is closed in the metric compact space Y , it iscompact.

This is the first time we need to verify that H(ϕ, ε) is not empty, andneither is XN : a simple check shows that for every ϕ, ε > 0 and n ∈ Z, thesingleton n belongs to H(ϕ, ε), and so XN 6= ∅ for all N ≥ 1. The inter-section X = ∩∞N=1XN is an intersection of a decreasing family of compactnon-empty sets, and so X is compact, and it’s members represent subsetsΛ ⊂ Z with the property that ΛN ∈ H(ϕ, ε) for every N . From lemma3.2.2 3, we see that Λ ∈ X if and only if Λ ∈ H(ϕ, ε), when we identify asusual a subset Λ ⊂ Z with an element of Y . Moreover, from lemma 3.2.2 1we see that TX = X, that is, X is translation-invariant. Denote by 0 theelement x ∈ Y all of whose entries are zeros. At this point of the discussionit is not yet clear whether X ∼= H(ϕ, ε) contains anything else than 0 orsingletons n. However, a theorem by Bourgain-Tzafriri ensures that thereexists 0 < c = c(ε, ϕ) such that if k ≥ 1

c then we can find σ ⊂ [1, k] with|σ| ≥ ck and σ ∈ H(ϕ, ε). This fact, together with a result by Rusza, enableus to prove the following result.


Theorem 3.2.1. Let ϕ ∈ L∞(0, 2π) be real-valued, and ε > 0. There existsa set Λ ⊂ Z with positive density such that ||PΛ(Lϕ − ϕ(0)I)PΛ|| < ε.

Proof. The family H(ϕ, ε) is homogeneous in the following sense:

1. if Λ ∈ H(ϕ, ε) and τ ⊂ Λ, then τ ∈ H(ϕ, ε).

2. if Λ ∈ H(ϕ, ε) and k ∈ Z then Λ + k ∈ H(ϕ, ε).

(1),(2) follow at once from lemma 3.2..2. According to a result by Rusza, ifH is a homogeneous system of finite subsets of Z, there exists and infinitesubset Λ ⊂ N such that:

(a) Every finite subset of Λ is in H.

(b) The density of Λ is given by: densΛ = limN→∞max |τ∩[1,N ]N : τ ∈ H

Applying 1. and 2. above to the finite subsets of H(ϕ, ε), we get fromRusza’s theorem that there exists a set Λ ⊂ N so that Λ ∈ H(ϕ, ε) by part(a), and densΛ ≥ c(ε, ϕ) by part (b) and Bourgain-Tzafriri’s theorem.

Corrolary 3.2.1. Let ϕ ∈ L∞(0, 2π) be real-valued, ϕ(0) 6= 0. There existsc = c(ϕ) > 0 and a subset Λ ⊂ N with positive density such that∣∣∣∣∫ 2π

0ϕ(t)|f(t)|2 dt

∣∣∣∣ ≥ c∫ 2π

0|f(t)|2 dt

for every f ∈ ImPΛ, ||f || ≤ 1.

Proof. Suppose on the contrary that for every δ > 0 and every Λ ⊂ N withpositive density we could find 0 6= f ∈ ImPΛ such that∣∣∣∣∫ 2π

0ϕ(t)|f(t)|2 dt

∣∣∣∣ < δ

∫ 2π

0|f(t)|2 dt

Take ε = |ϕ(0)|2 . From theorem 3.2.1 we have a set Λ1 ⊂ N with positive

density for which

|ϕ(0)|2

> ||PΛ1(Lϕ−ϕ(0)I)PΛ1 || = sup|〈ϕf, f〉−ϕ(0)||f ||2| : f ∈ ImPΛ1 , ||f || ≤ 1

On the other hand, by assumption we have g ∈ ImPΛ1 , ||g|| = 1 such that

|〈ϕg, g〉 =

∣∣∣∣∫ 2π

0ϕ(t)|g(t)|2 dt

∣∣∣∣ < |ϕ(0)|3

∫ 2π

0|g(t)|2 dt =

|ϕ(0)|3

combining these gives:

|ϕ(0)|2

< |〈ϕg, g〉| < |ϕ(0)|3

a contradiction.


Remark 3.2.1. If B ⊂ (0, 2π) is a Borel set with positive Lebesgue measure,then in the previous corollary we may take ϕ = χB, the indicator function ofB. We have χB(0) = m(B), and obtain a slightly weaker version of Theorem2.2 in [4]. The difference is that in their theorem the constant c is absolute,while we get a constant depending on the set B.

We saw that H(ϕ, ε) contains sets of positive density, and at this pointwe do not know if any other infinite sets belong to H(ϕ, ε). Let’s show thatthere are infinite sets, (although of density zero), that belong to H(ϕ, ε).We need a lemma.

Lemma 3.2.3. Let Λ ∈ Z and Λ−Λ the set of differences m−n : m,n ∈ Λ.Let DΛ = Λ− Λ\0. The for every real-valued ϕ ∈ L∞:

||PΛ(Lϕ − ϕ(0)I)PΛ|| ≤ ||PDΛϕ||

Proof. By lemma 2.1.1 it suffices to check the inequality on finite subsets ofΛ. Let σ ⊂ Λ be finite, then:

||Pσ(Lϕ − ϕ(0)I)Pσ|| =∑|〈ϕf, f〉 − ϕ(0)||f ||2| : ||f || ≤ 1, f ∈ ImPσ

Given f ∈ ImPσ, ||f || ≤ 1, we can write f =∑

n∈σ cnen,∑

n∈σ |cn|2 ≤ 1, andwe have:∣∣〈ϕf, f〉 − ϕ(0)||f ||2

∣∣ =

∣∣∣∣∣ ∑n,m∈σ

cncmϕ(m− n)− ϕ(0)∑n∈σ|cn|2

∣∣∣∣∣=

∣∣∣∣∣∣∑n6=m

cncmϕ(m− n)

∣∣∣∣∣∣<

( ∑m,n∈σ

|cn|2|cm|2)1/2

∑m 6=n|ϕ(m− n)|2

1/2

≤

(∑n∈σ|cn|2

)21/2

· ||PDΛ||

≤ ||PDΛ||

We now introduce a class of subsets of N all belonging to H(ϕ, ε), forany ε > 0 and real-valued ϕ ∈ L∞(0, 2π).

Definition 3.2.1. We say that Λ = ni∞i=1 ⊂ N is of type (k, l) if thereexist natural numbers k, l so that

(i) The gap (i.e., difference) between any two consecutive members of Λis bounded below by k: ni+1 − ni ≥ k for all i.


(ii) No natural number can be written as a difference of elements of Λ inmore than l ways.

Example 3.2.1. For each k, define a sequence by n0 = 0, n1 = k, and nj+1 =nj + k + j. That is, 0, k, 2k + 1, 3k + 3, 4k + 6 etc. This is an example of aset of type (k, 1), as can be easily checked.

To obtain a set of type (k, l) for l > 1, start with the previously definedsequence nj and consider lnjj≥0. We have lnj+1− lnj = l(k+j). Betweenlnj and lnj+1 insert the sequence

lnj < lnj+(nj+1−nj) < lnj+2(nj+1−nj) < . . . < lnj+l(nj+1−nj) = lnj+1

The sequence thus obtained is of type (k, l).

A sequence of type (k, l) must have zero density, because by a theoremof Szemeredi, if Λ ⊂ N has positive density then for every l Λ contains anarithmetic progression n1, n1 + d, . . . , n1 + (l − 1)d, and in particular d canbe represented as a difference of members of Λ in at least l ways.

Proposition 3.2.1. Let ϕ ∈ L∞(0, 2π) be real-valued, and ε > 0. The forevery l ∈ N there exists a natural number k = k(ε, ϕ, l) such that H(ϕ, ε)contains every set Λ ⊂ N of type (k, l).

Proof. Choose k = k(ε, ϕ, l) so that(∑

|j|≥k |ϕ(j)|2)1/2

≤ εl−1/2. Let Λ ⊂ Nbe a set of type (k, l). Let DΛ be as in lemma 3.2.3. Then:∑

j∈DΛ

|ϕ(j)|21/2

= ||PDΛϕ|| ≤

l ∑|j|≥k

|ϕ(j)|21/2

< ε

and the result follows from lemma 3.2.3.

Since every finite arithmetic progression of length k and common dif-ference d is of type (k, d), proposition 3.2.1 shows that H(ϕ, ε) containsarithmetic progressions of arbitrary length. We could also have deducedthat from the fact that H(ϕ, ε) contains a set of positive density.

3.2.1 The paving property and syndetic sets

Definition 3.2.2. A subset Λ ⊂ Z is called syndetic if Z can be covered bya finite number of translations of Λ, that is, if there exists l ∈ N such that

Z = Λ + k1 ∪ Λ + k2 · · · ∪ Λ + kl

for some k1, . . . , kl ∈ Z.


Clearly a syndetic set has bounded gaps. Syndetic subsets of N areprecisely those whose gaps are bounded, but in Z we need to distinguishbetween the property of bounded gaps and being syndetic, as 2n∞n=1 ⊂ Zhas bounded gaps, but no finite number of it’s translations cover Z.

Since H(ϕ, ε) is translation-invariant, if we can find a syndetic set Λ ∈H(ϕ, ε) then Lϕ has the paving property, because if for some l and integers

k1, . . . , kl we have Z =⋃li=1 Λ + ki then we can pass to a disjoint covering

of Z by choosing appropriately subsets σi ⊂ Λ + ki, without increasing thenorms ||Pσi(Lϕ− ϕ(0)I)Pσi ||. As it turns out, the existence of a syndetic setin H(ϕ, ε) for every ε > 0 is also necessary for Lϕ (ϕ real-valued) to havethe paving property. This will follow from the following result (for a proof,see [8], 1.23).

Theorem 3.2.2. Given a partition of Z into a finite number of disjoint setsΛ1, . . . ,Λm, there exists an l ∈ N such that one of the sets, say Λj, containsarbitrarily long finite sequences whose gaps are bounded by l. That is, forevery k ∈ N we can find n1 < n2 < · · · < nk in Λj for which ni+1 − ni ≤ l.

A set with the property described in the theorem is called piecewisesyndetic ([8], 1.11).

Lemma 3.2.4. Let X ⊂ Y = 0, 1Z a compact set. Assume there is anx ∈ X so that n|x(n) = 1 is piecewise syndetic. Then X contains a ysuch that n|y(n) = 1 has bounded gaps, and if X is translation-invariant,y may be chosen so that n|y(n) = 1 is syndetic.

Proof. Under the assumption, there exists some l ∈ N so that for each kthere exists an element xk ∈ X so that n|xk(n) = 1 corresponds to asubset n1 < n2 < · · ·nk contained in n|x(n) = 1 for which ni+1 − ni ≤ lfor all i = 1, 2, . . . , k − 1. As X is compact, the sequence xk∞k=1 hasan accumulation point y ∈ X. Put Λ = n|y(n) = 1. Let n1 < n2

be consecutive members of Λ. Take δ = 1max|n1,|n2|+1 , then for some k

d(xk, y) < δ. By the choice of δ, xk(i) = yk(i) for every n1 ≤ i ≤ n2. Inparticular xk(n2) = xk(n1) = 1 and so n1, n2 both belong to n|xk(n) = 1,and n1 < n2 are also consecutive in this set, in which the gaps are boundedby l. Thus n2 − n1 ≤ l. This shows that Λ has gaps bounded by l. Now,if X is translation-invariant, then for every k > 0 we can find xk ∈ X suchthat xk(n) = 1 if and only if |n| ≤ [k/2] (where [m] is the integer part ofm). In this way we will have obtained arbitrarily large finite sets, symmetricaround zero, whose gaps are bounded by l, all belonging to n|xk(n) = 1.An accumulation point of such a collection will be syndetic.

So we have our main result:

Theorem 3.2.3. Let ϕ ∈ L∞(0, 2π) real-valued. The Laurent operator Lϕhas the paving property if and only if for every ε > 0, there exists a syndetic


set Λ ⊂ Z such that||PΛ(Lϕ − ϕ(0)I)PΛ|| < ε

(”syndetic” may be replaced with ”piecewise syndetic”)

Although every set of positive density contains arbitrarily long arithmeticprogressions, the common difference of these is not controlled. Consequently(and unfortunately), not every set of positive density (which we have shownto be included in H(ϕ, ε)) is piecewise syndetic:

Example 3.2.2. Consider the set

Λ =

∞⋃k=1

∞⋃n=1

[n · 10k, n · 10k + k)

For N ∈ N,

|Λ ∩ [1, N ]| =M(N)−1∑k=1

N

10k−R(N)

where the order of magnitude of M(N) is the number of digits of N in it’s

decimal expansion, and R(N)N → 0 as N →∞. dividing both sides by N and

letting N →∞ we get that the density of Λ is∑∞

k=1 10−k = 1/9. Considerthe complement set Λc. It’s density is 8/9 but it is not piecewise syndetic.In fact, for each l ∈ N, and every sequence n1 < n2 < · · · < n102l in Λc, wewill find consecutive nj , nj+1 such that nj+1 − nj > l. The reason is that ifwe choose 102l members of Λc some of them must be greater than 102l, butnone of them belong to [102l, 102l + 2l), whose length is 2l. As a result, thegap between two such elements must exceed l.

We shall end this chpater with a short discussion about the case whenϕ ∈ L∞(0, 2π) is continuous. It is known that in this case for every ε > 0there exists a uniform paving, that is, a paving with arithmetic progressions,of the operator Lϕ. ([12]). Actually (HKW) showed that every Lϕ withϕ Riemann-integrable is (uniformly) paveable. We shall give here a shortconstructive proof for the continuous case, by showing that there exists auniform paving for a class of operators whose closure contains all operatorswith continuous symbols. The fact that the set of paveable operators formsa closed subspace of the Banach space B(l2) (see [12]) plays a role here.

Proposition 3.2.2. Let ϕ ∈ C(0, 2π), and ε. There exists d ∈ N suchthat H(ϕ, ε) contains the arithmetic progression dkk∈Z, and so Lϕ has auniform paving.

Proof. For rational 0 ≤ p < q ≤, let χpq denote the indicator function of theinterval [2πp, 2πq]. Then

χpq(n) =i

2πne−in2πq(1− ein2π(p−q))


Let d be the first natural number for which d(p−q) ∈ N. Then for all k ∈ Z,χpq(kd) = 0. Put Λ = kdk∈Z. Then the difference-set Λ − Λ coincideswith Λ. From lemma 3.2.3:

||PΛ(Lχpq − (q − p)2πI)PΛ|| ≤ ||PDΛχpq|| = 0

Λ is an arithmetic progression, so every χpq has a uniform paving. Givena continuous ϕ ∈ C(0, 2π), uniform continuity of ϕ ensures that for everyε > 0 we can find rational points 0 = p0 < p1 < · · · pl so that pi − pi−1 <δ(ε), and for every 2πpi−1 ≤ s, t ≤ 2πpi |ϕ(s) − ϕ(t)| < ε. Choose pointsξi ∈ [2πpi−1, 2πpi]. Then for all 0 ≤ t ≤ 2π:∣∣∣∣∣

l∑i=0

ϕ(ξi)χpi−1pi(t)− ϕ(t)

∣∣∣∣∣ = |ϕ(ξi)− ϕ(t)| < ε

because t ∈ [2πpi−1, pi] for some 0 ≤ i ≤ l. Thus, the closure in L∞(0, 2π)of the finite linear combinations of the indicator functions χpq contains thecontinuous functions, and the proposition is proved.

Chapter 4

Equivalent Formulations

4.1 Six equivalent formulations

Theorem 4.1.1. The following conditions are equivalent.

Uniqueness of Extension. Every pure state f ∈ P (D) has a uniqueextension to a pure state F ∈ P (B(l2)).

Paving property. Every operator T ∈ B(l2) has the paving property:for every ε > 0 there is a partition of N into m = m(ε, T ) disjointsubsets σ1, . . . , σm such that

||Pσj (T − E(T ))Pσj || < ε||T ||, j = 1, 2, . . . ,m

Relative Dixmier property. Every operator T ∈ B(l2) has therelative Dixmier property, i.e., the set

K(T ) = convDTD∗ : D diagonal unitary

contains the diagonal of T .

Local paving property. For every ε there exists m = m(ε) suchthat for every n and every T : ln2 → ln2 there exists a partition of1, 2, . . . , n into m disjoint subsets σj so that

||Pσj (T − E(T ))Pσj || < ε||T ||, j = 1, 2, . . . ,m

Restricted invertibility. There exists a real number c > 0 andm ∈ N so that for every n and every T : ln2 → ln2 satisfying ||Tei|| = 1∀i,there is a partition of 1, 2, . . . , n into m disjoint sets σj, so that forevery choice of scalars ai:

||∑i∈σj

aiTei|| ≥ c

∑i∈σj

|ai|21/2

(j = 1, 2, . . . ,m)

33

34 CHAPTER 4. EQUIVALENT FORMULATIONS

Diagonalization property. For every operator T ∈ B(l2) and everypure state f ∈ P (D):

βf (T ) = inf||Pσ(T − E(T ))Pσ|| : σ ∈ Uf = 0

where Uf is the ultra-filter corresponding to f .

Kadison and Singer proved the equivalence between the paving propertyand the uniqueness of extension in [14] (lemma 5). The paving propertyhas attracted considerable attention. Kadison and Singer proved that everyoperator represented by a permutation matrix (i.e. every row and every col-umn contains exactly one 1 and the rest are zeros) has the paving property.([14], Th. 3). Then Gregson [10] showed that opertaors all of whose matrixentries are non-negative have the paving property too. Berman, Halpern,Kaftal and Weiss independently obtained the same result ([2]). As was dis-cussed in chapter 3, Halpern Kaftal and Weiss showed that the operator Lϕthat multiplies each f ∈ L2[0, 2π] by ϕ ∈ L∞(0, 2π) (a Laurent operator)has the paving property if ϕ is Riemann integrable. Bourgain and Tzafririshowed that there exists a sub-algebra of the algebra of Laurent operatrosall of whose members have the paving property. The corresponding multi-pliers ϕ ∈ L∞(0, 2π) are characterized by a certain decay condition on theirFourier coefficients ([5]). The question whether every Laurent operator hasthe paving property is still open.

Using characterization of singular states, it can be shown that everycompact operator S ∈ B(l2(N)) has the paving property: If S is compact andρ1, ρ2 are two extensions of a singular state ρ, then ρ1, ρ2 are also singular,and so ρ1(S) = ρ2(S) = 0. The diagonal of a compact operator E(S) is alsoa compact operator, because limn→∞〈Sen, en〉 = 0, and so ρ1(S) = ρ2(S) =ρE(S). From here it follows that ρ1, ρ2, ρE coincide on the two-sided idealof the compact operators, and that S has the paving property follows fromKadison and Singer’s proof of the equivalence between the paving propertyand the uniqueness of extension.

All these problems about operators in B(l2(N)) can be localized to ln2 .If T : l22 → ln2 then T can be trivially ”paved” by the projections Pi =ei ⊗ ei where e1, . . . , en is the standard basis in ln2 . Such pavings giveno information as n → ∞. The question here is whether the local pavingproperty holds. The only result in this direction was obtained by Bourgainand Tzafriri in [3]. They proved that for every ε > 0 and each M > 0there is a constant c = c(M, ε) such that for each n ≥ 1/c and every linearS : ln2 → ln2 with zeros on the diagonal and ||S|| ≤ M , there exists a subsetσ ⊂ 1, . . . , n whose size is at least nc, such that ||PσSPσ|| < ε.

in [12] Halpern Kaftal and Weiss proved that if B ⊂ (0, 2π) is anopen set such that the boundary ∂B has positive Lebesgue measure, andm(B) > 1/2, then for every arithmetic progression Λ ⊂ Z one has ||PΛ(LχB−m(B)I)PΛ|| ≥ m(B). (Here m(B) is the Lebesgue measure and χB is the

4.2. UNIQUENESS OF EXTENSION⇐⇒DIAGONALIZATION PROPERTY35

characteristic function of B). Consequently, the authors suggested the pos-sibility of a negative solution to the paving property, but Bourgain andTzafriri came up with an example of an open set V ⊂ (0, 2π) whose bound-ary has full measure yet the Laurent operator LχV does have the pavingproperty.

The theorem is proved by showing the following equivalences:

1. uniqueness of extension =⇒ paving property.

2. paving property =⇒ relative Dixmier property.

3. relative Dixmier property =⇒ uniqueness of extension.

4. uniqueness of extension ⇐⇒ diagonalization property.

5. paving property ⇐⇒ local paving property.

6. local paving property ⇐⇒ restricted invertibility.

4.2 uniqueness of extension ⇐⇒ diagonalizationproperty

This was proved in theorem 2.3.1.

4.3 uniqueness of extension =⇒ paving property

This was proved by Kadison and Singer in [14]. If every f ∈ P (D) has aunique extension to a pure state of B(l2), then a careful examination of theproof of theorem 2.3.1 shows that for each f ∈ P (D) we can find a closed andopen set W ⊂ P (D) containing f so that if PW is the projection correspond-ing to the indicator χW , then ||PW (T−E(T ))PW || < ε, where ε, T are apriorigiven. As P (D) is compact, we can find a cover of P (D) by W1, . . . ,Wm,where m depends on ε, T only, such that ||PWj (T − E(T ))PWj || < ε for all1 ≤ j ≤ m. Consider the sets

σj = n ∈ N|〈PWjen, en〉 = 1

then⋃σj = N, because

⋃Wj = P (D) implies

∑mj=1 PWj = I. Passing to a

disjoint covering of N we obtain the paving property.

4.4 paving property =⇒ relative Dixmier property

Let T ∈ B(l2) and ε > 0. If σ ⊂ N and σc is the complement of σ, thenU = Pσ − Pσc is a diagonal unitary operator, and:

1

2[T + UTU∗] = PσTPσc + PσcTPσ (4.1)


Thus, if for ε, T ) there are m = m(ε, T ) disjoint subsets σ1, . . . , σm for which||Pσj [T − E(T )]Pσj || < ε||T ||, then from (4.1) we can write:

m∑j=1

Pσj [T − E(T )]Pσj =m∑j=1

αjUj [T − E(T )]U∗j (4.2)

where∑m

j=1 αj = 1, αj ≥ 0 and Uj is unitary and diagonal, for each 1 ≤j ≤ m. From (4.2) we have:

||m∑j=1

||Pσj (T−E(T ))Pσj |||| = max1≤j≤m

||||Pσj (T−E(T ))Pσj |||| = ||m∑j=1

αjUj(T−E(T ))U∗j ||

(4.3)and (4.3) shows that for apriori given ε, T , we have found an element in theset

conv U [T − E(T )]U∗|U unitary and diagonal

whose norm is at most ε||T ||. As ε > 0 is arbitrary, this means that theclosure of this convex hull contains the zero operator, which equivalent toK(T ) containing the diagonal of T . Thus the paving property implies therelative Dixmier property. To show that relative Dixmier property =⇒uniqueness of extension we use proposition 2.3.1. If f ∈ P (D) and F isan extension of f , then for unitary and diagonal U :

F (U(T − E(T ))U∗ = F (U)F (U∗)F (T − E(T )) = F (T − E(T ))

so if S ∈ K(T−E(T )) then F (S) = F (T−E(T )). Since the relative dixmierproperty implies that 0 ∈ K(T − E(T )), we have: F (T − E(T )) = F (T ) −F (E(T )) = 0, hence F (T ) = F (E(T )). In other words, the compositionF = f E is the only extension of f to a state of B(l2). From proposition(2.3.1)(3) we deduce that f is a unique extension to a pure state of B(l2).

4.5 paving property ⇐⇒ local paving property

paving property =⇒ local paving property Take τj = σj ∩ [1, n].

local paving property =⇒ paving property : In [2] the authorsdefine for each T ∈ B(l2(N)) the numbers

α(T ) = inf||∑

PmTPm|| :∑

Pm = I, Pi ⊥ Pj (i 6= j)

where the infimum is taken over finite sums of mutually orthogonal projec-tions whose sum is the identity. Similarly they denote αk(T ) to be the sameinfimum taken over sums of k projections. They prove (prop. 2.6) that forevery T and k: αk(T ) = limn→∞ αk((P[1,n]TP[1,n]), where P[1,n] is the orthog-onal projection onto the first n unit vectors. Now let T ∈ B(l2(N)), ε > 0 be

4.6. LOCAL PAVING PROPERTY =⇒ RESTRICTED INVERTIBILITY37

given, with ||T || ≤ 1. Let m = m(ε) as in the local paving property. Let n besufficiently large so that |αm(T − E(T )) − αm(P[1,n](T − E(T ))P[1,n])| < ε,and since in the expression P[1,n](T−E(T ))P[1,n]) we may replace (T−E(T ))by it’s restriction to ln2 , then by the local paving property we deduce that

αm(P[1,n](T − E(T ))P[1,n]) < ε||T|ln2 || ≤ ε||T || ≤ ε

and so αm(T − E(T )) < 2ε.

4.6 local paving property =⇒ restricted invertibil-ity

Assume that restricted invertibility does not hold. Then for every c > 0there is some n = nc ∈ N and a linear map T : ln2 → ln2 with ||Tei|| = 1 forall i, such that whenever σ1, . . . , σm is a partition of [1, nc] we can find some1 ≤ j ≤ m and a vector x0 =

∑i∈σj aiei such that

||Tx0|| = ||∑i∈σj

aiTei|| < c

∑i∈σj

|ai|21/2

= c||x0|| (4.4)

from which it follows that:

〈T ∗Tx0, x0〉 < c2||x0||2 (4.5)

With no loss of generality, we may assume ||x0|| ≤ 1. The assumption ||Tei|| =1 implies 〈T ∗Tei, ei〉 = 1 for each i, and so the diagonal of T ∗T is the identity

matrix. Choose c so that c <√

23 . Take ε = c2/2 and use the local paving

property for the operator T ∗T and n = nc. There is some partition τ1, . . . , τlof 1, . . . , n so that

||Pτj (T ∗T − I)Pτj || < ε ∀1 ≤ j ≤ l (4.6)

Since T ∗T is self-adjoint, so is Pτj (T∗T − I)Pτj for each j, and so the norm

can be computed as:

||Pτj (T ∗T − I)Pτj || = sup|〈Pτj (T ∗T − I)Pτjx, x〉| : x ∈ ln2 , ||x|| ≤ 1= sup|〈T ∗Ty, y〉 − ||y||2| : y ∈ Pτj ln2 , ||y|| ≤ 1

by our contradiction-assumption and the last equation:

〈T ∗Tx0, x0〉 < c2||x0||2 (4.7)

||x0||2 − c2/2 < 〈T ∗Tx0, x0〉 < ||x0||2 + c2/2 (4.8)

combining (4.7) and (4.8) we get

0 < ||x0||2(c2 − 1) + c2/2 ≤ 3c2/2− 1

contradicting the choice of c. Thus the local paving property implies re-stricted invertibility.


4.7 restricted invertibility =⇒ local paving prop-erty

We shall need the following result, proved in [3]:

Theorem 4.7.1. Let aij1≤i,j≤n be a matrix satisfying:

(i) aij ≥ 0 for all 1 ≤ i, j ≤ n.

(ii)∑n

j=1 aij ≤ 1 for all 1 ≤ i ≤ n.

Then for every k there exists a partition σ1, . . . , σk of 1, 2, . . . , n into kdisjoint subsets so that for each 1 ≤ l ≤ k:∑

j∈σl

aij ≤2

ki ∈ σl

Let ε > 0 and S : ln2 → ln2 a linear map with diagonal zero. Let sij bethe matrix representing S with respect to the standard basis. Then:

(1 ≤ i ≤ n)n∑j=1

|sij |2 = ||Sei||2 ≤ ||S||2

We shall apply theorem 4.0.5 to the matrix aij =|sij |2||S||2 . Take an integer

k > 2 such that 2||S||2k < ε2, and take a partition σ1, . . . , σk of 1, . . . , n

such that

(i ∈ σk, 1 ≤ l ≤ k)∑j∈σl

aij ≤2

k

That is:

(i ∈ σk, 1 ≤ l ≤ k) ||PσlSei||2 =

∑j∈σl

|sij |2 ≤2||S||2

k(4.9)

For 1 ≤ i ≤ n and 1 ≤ l ≤ k put

xli =1

||S||

√k

2PσlSei (4.10)

Then we have:

(a) ||xli|| ≤ 1 for each i ∈ σl and for each 1 ≤ l ≤ k.

(b) For every choice of cii∈σl :

||∑i∈σl

cixli|| ≤

√k

2

(∑|ci|2

)1/2(4.11)

4.7. RESTRICTED INVERTIBILITY =⇒ LOCAL PAVING PROPERTY39

(a) follows from 4.9 and 4.10, and (b) follows because for every u =∑ni=1 ciei:

||PσlSPσlu|| ≤ ||PσlS||||Pσlu|| ≤ ||S||

∑i∈σl

|ci|21/2

=⇒ ||PσlS∑i∈σl

ciei|| ≤ ||S||

∑i∈σl

|ci|21/2

=⇒ ||∑i∈σl

ciPσlSei|| ≤ ||S||

∑i∈σl

|ci|21/2

=⇒ ||∑i∈σl

ci||S||√

2

kxli|| ≤ ||S||

∑i∈σl

|ci|21/2

and this is equivalent to (b).The idea is now to partition the sets σ1, . . . , σl recursively, each time

reducing the number√k/2 appearing in (b). In the first step, we define k

new matrices B1, . . . , Bk by:

(1 ≤ l ≤ k), (M =√k/2) : (bij)

l =M2

M2 − 1δij −

〈xli, xlj〉M2 − 1

, i, j ∈ σl

The given u in the range of the projection Pσl , we have 〈Blu, u〉 ≥ 0, foreach 1 ≤ l ≤ k. Indeed, if u =

∑i∈σl ciei, then

(1 ≤ l ≤ k) 〈Blu, u〉 =∑i,j∈σl

(bij)lcicj

=1

M2 − 1

∑i,j∈σl

[M2δij − 〈xli, xlj〉]cicj

=M2

M2 − 1

∑i∈σl

|ci|2 −||∑

i∈σl cixlj ||

M2 − 1≥ 0

the last inequality being true thanks to (b). Since the Bl’s are positiveoperators, they have square roots, i.e., S2

l = Bl. Put zli = Slei, for i ∈ σl.Then

〈zlj , zli〉 = 〈Slej , Slei〉 = 〈Blej , ei〉 = (bij)l

hence,

||∑i∈σl

cizli||2 =

∑i,j∈σl

〈cjzlj , cizli〉 (4.12)


=∑i,j

(bij)lcjci (4.13)

=

M2∑i∈σl

|ci|2 − ||∑i∈σl

cizli||2 /(M2 − 1) (4.14)

as we computed before. (4.12–4.14) hold for every 1 ≤ l ≤ k and everychoice of scalars cii∈σl . If we take in (4.12–4.14) ci = 1 for a single indexi and ci = 0 for all the rest, we get:

||zli||2 =M2 − ||xli||2

M2 − 1≥ M2 − 1

M2 − 1= 1 (4.15)

because ||xli|| ≤ 1 for all 1 ≤ l ≤ k and i ∈ σl. Also, from (b) we have:

1

M2 − 1

M∑i∈σl

|ci|21/2

− ||∑

cixli||

M

∑i∈σl

|ci|21/2

+ ||∑

cixli||

≤ 1

M2 − 1

2M

∑i∈σl

|ci|21/2

2

=4M2

M2 − 1

∑i∈σl

|ci|2

Combining this with (4.12-4.14) gives:

||∑i∈σl

cizli|| ≤

2M√M2 − 1

∑i∈σl

|ci|21/2

=2k1/2

(k − 2)1/2

∑i∈σl

|ci|21/2

(4.16)

(recall that M =√k/2). Consider the operators Tl : l

|σl|2 → l

|σl|2 defined by

Tlei =zli||zli||

. One has ||Tlei|| = 1 and by (4.15), (4.16) also ||Tl|| ≤ 2k1/2

(k−2)1/2 .

At this point we use restricted invertibility. We get a constant c > 0 andpartitions ηl1, · · · , ηlm of each one of the sets σl, 1 ≤ l ≤ k such that for everychoice of scalars ci:

(1 ≤ l ≤ k, 1 ≤ t ≤ m) : ||∑i∈ηlt

cizli|| ≥ c

∑i∈ηlt

|ci|21/2

From (4.12–4.14) we get:

1

M2 − 1

M2∑i∈ηlt

|ci|2 − ||∑i∈ηlt

cixli||2 ≥ c2

∑i∈ηlt

|ci|2

4.7. RESTRICTED INVERTIBILITY =⇒ LOCAL PAVING PROPERTY41

hence

||∑i∈ηlt

cixli||2 ≤ M2

∑i∈ηlt

|ci|2 − (M2 − 1)c2∑i∈ηlt

|ci|2

= (M2 − (M2 − 1)c2)∑i∈ηlt

|ci|2

for all 1 ≤ t ≤ m, 1 ≤ l ≤ k. Denote M1 = (M2 − (M2 − 1)c2)1/2. Then:

||∑i∈ηlt

cixli|| ≤M1

∑i∈ηlt

|ci|21/2

for all 1 ≤ t ≤ m, 1 ≤ l ≤ k, and every choice of scalars ci, i ∈ ηlt. Wenow iterate the same process for the new partition. We will have obtaineda further refinement of the partition, and a number M2 given by:

M22 = (1 + (M1 − 1)(1− c2)) = 1 + (M2 − 1)(1− c2)2

Repeating this sufficiently times we shall get:

(M2 − 1)(1− c2)N < 1

and a refinement to a partition Λ1, . . . ,Λm(ε) so that

||∑i∈Λj

cixji || ≤ (M2 − 1)(1− c2)N

∑i∈Λj

|ci|21/2

and so

||∑i∈Λj

ci1

||S||

√k

2PΛjSei|| ≤ (M2 − 1)(1− c2)N

∑i∈Λj

|ci|21/2

therefore:

||PΛjS∑i∈Λj

ciei|| ≤ ||S||√

2

k

∑i∈Λj

|ci|21/2

and since we took 2||S||2k < ε2, we get for all x:

||PΛjSPΛjx|| ≤ ε||PΛjx|| ≤ ε||x||

so ||PΛjSPΛj || ≤ ε for each 1 ≤ j ≤ m(ε). This completes the proof that re-stricted invertbility implies the local paving property, and of the equivalencetheorem.

Bibliography

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[2] Berman, K., Halpern, H., Kaftal, V., Weiss, G., Matrix norm inequal-ities and the relative Dixmier property. Integral Eq. and Oper. Theory11 (1988), 1–48.

[3] Bourgain, J., Tzafriri, L., Restricted invertibility of matrices and appli-cations. Analysis at Urbana, Vol. II (Urbana, IL, 1986–1987), LondonMath. Soc. Lecture Note 138, Cambridge Univ. Press (1989)

[4] Bourgain, J., Tzafriri, L., Invertibility of ”large” submatrices with ap-plications to the geometry of Banach spaces and harmonic analysis.Israel Journal of Mathematics 57 (1987), 137–224.

[5] Bourgain, J., Tzafriri, L., On a problem of Kadison and Singer. Journalfur die reine und angewandte mathematik 420 (1991) 1–44.

[6] Dirac, P.A.M., The principles of Quantum Mechanics . Oxford Univer-sity Press (1958)

[7] Fano, G., Mathematical methods of Quantum Mechanics. McGraw-Hill(1971)

[8] Furstenberg, H., Recurrence in Ergodic theory and combinatorial num-ber theory. Princeton University Press NJ (1981).

[9] Glimm, J., A Stone-Weierstrass theorem for C∗ algebras. Ann. Math.72 (1960) 216–243.

[10] Gregson, K., Extension of pure states of C∗-algebras. Ph.D Disserta-tion, Univ. of Aberdeen. (1986).

[11] Halmos, P.R., A Hilbert space problem book. Van Nostrand, (1967).

[12] Halpern, H., Kaftal, V., Weiss, G., Matrix pavings and Laurent opera-tors. J. Oper. Theory 16 (1986) 335–374.

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44 BIBLIOGRAPHY

[13] Kadison, R., Ringrose, J., Fundamentals of the theory of operator al-gebras Vol. II AP (1986) Vol. I (1983)

[14] Kadison, R., Singer, I. Extensions of pure states. Amer. J. Math. 81(1959) 547–564.

[15] Rudin, W., Functional Analysis. Tata McGraw-Hill, (1973).

lonke msc thesis

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