c -algebras a dissertation submitted tocj... · 2019. 2. 13. · a dissertation submitted to the...

Callias-type Operators in C∗-algebras

by Simone Cecchini

B.S. in Philosophy and M.S. in Mathematics, University of Rome “La Sapienza”

A dissertation submitted to

The Faculty ofthe College of Science ofNortheastern University

in partial fulfillment of the requirementsfor the Degree of Doctor of Philosophy

April, 2017

Dissertation directed by

Maxim BravermanProfessor of Mathematics

This disseration is dedicated to my fiancé Khadidiatou, to my parents, Felice and Tilde,

and to my brother Federico.

ii

Acknowledgments

I would like to express my deepest thanks to my advisor, Professor Maxim Braverman, for

having introduced me to many fascinating mathematical topics, for his continuous encour-

agement, and for having shared his knowledge during many discussions.

I also would like to thank Professor Robert McOwen, Professor Petar Topalov, and

Professor Gideon Maschler for being so kind as to serve in my dissertation committee.

In addition, I would like to express many thanks to my friends and colleagues at North-

eastern, Chen, Gouri, Emre, Reuven, José, Andras, Floran, Neranga, Brian, Boris, Saif,

Tong, Rahul, Nate, Ivan, Whitney, Jonier, Monika, Edgar, and many others with whom I

shared pleasant memories at the department. Thanks to Beth, Donika, Ian and Chantal for

various logistical support over the years.

Simone Cecchini

Northeastern University

April 2017

iii

Abstract of Dissertation

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a

Schrödinger-type operator with a potential uniformly positive outside of a compact set. We

develop the theory of Callias-type operators twisted with Hilbert C∗-module bundles and

prove an index theorem for such operators. As an application, we derive an obstruction to

the existence of complete Riemannian metrics of positive scalar curvature on noncompact

spin manifolds in terms of closed submanifolds of codimension-one. In particular, when

N is a closed even dimensional spin manifold, we show that if the cylinder N × R carries

a complete metric of positive scalar curvature, then the (complex) Rosenberg index on N

must vanish.

iv

Table of Contents

Acknowledgments iii

Abstract of Dissertation iv

Table of Contents v

Disclaimer vii

Chapter 1 Introduction 1

Chapter 2 The index of twisted Callias-type operators 6

2.1 Twisted Dirac-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Hilbert A-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Hilbert A-bundles and connections . . . . . . . . . . . . . . . . . . . 9

2.1.3 Dirac-type operators twisted with Hilbert bundles . . . . . . . . . . . 11

2.2 Schrödinger-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Unbounded operators on Hilbert A-modules . . . . . . . . . . . . . . 12

2.2.2 Operators of Schrödinger-type . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Regularity of H∗µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.4 Symmetricity of H∗µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Twisted Callias-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Twisted operators of Callias-type . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Bounded Kasparov modules . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

v

2.3.4 A-index of twisted Dirac-type operators: Bunke’s approach . . . . . . 21

2.3.5 Invertibility of B2V + f . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.6 A-index of twisted Callias-type operators . . . . . . . . . . . . . . . . 25

Chapter 3 Reduction to the cylinder 27

3.1 A Callias-type theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Some properties of the index of twisted Callias-type operators . . . . . . . . 30

3.2.1 A sufficient condition for the vanishing of the A-index . . . . . . . . . 30

3.2.2 Proof of Proposition 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.3 A compact perturbation of the potential . . . . . . . . . . . . . . . . 32


3.2.5 A family of operators invertible at infinity . . . . . . . . . . . . . . . 35


3.2.7 The opposite potential . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.8 The Fredholm picture . . . . . . . . . . . . . . . . . . . . . . . . . . 42


3.3 Reduction to the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 The model operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Bunke’s relative index theorem . . . . . . . . . . . . . . . . . . . . . 45

3.3.3 A manifold with the reversed orientation . . . . . . . . . . . . . . . . 47

3.3.4 Reduction to a manifold with cylindrical ends . . . . . . . . . . . . . 49

3.3.5 A perturbation of the connection on the cylindrical end . . . . . . . . 50

3.3.6 A perturbation of the potential on the cylindrical end . . . . . . . . . 52

3.3.7 Proof of Theorem 3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 4 The model operator 57

4.1 The model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.1 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 An unbounded Kasparov module . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Unbounded Kasparov modules . . . . . . . . . . . . . . . . . . . . . . 58

4.2.2 An alternative definition of the A-index of the model operator . . . . 59

vi

4.2.3 An A-compact operator . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.4 The localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.5 Idea of the proof of Theorem 4.2.2 . . . . . . . . . . . . . . . . . . . 61

4.2.6 Some estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.7 Proof of Theorem 4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.8 Equality of index classes . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.9 Proof of Theorem 4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Analysis on the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.1 Intersection product . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.2 Twisted Dirac operators on compact manifolds . . . . . . . . . . . . . 69

4.3.3 KK-theoretical version of Anghel’s theorem . . . . . . . . . . . . . . . 70

4.3.4 A-index of Twisted Callias-type operators . . . . . . . . . . . . . . . 70

4.3.5 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Chapter 5 Positive scalar curvature on noncompact manifolds 75

5.1 Higher Dirac obstructions on closed spin manifolds . . . . . . . . . . . . . . 75

5.2 The vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.1 Bochner-Lichnerowitz Formula . . . . . . . . . . . . . . . . . . . . . . 77

5.2.2 An admissible potential . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.3 Proof of Theorem 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Codimension-one obstructions to PSC on noncompact manifolds . . . . . . . 80

vii

Disclaimer

I hereby declare that the work in this thesis is that of the candidate alone and the use of all

material from other sources has been properly and fully acknowledged.

viii

Chapter 1

Introduction

An important topic in differential geometry in recent decades is whether a given smooth

manifold admits a Riemannian metric of positive scalar curvature. On closed spin manifolds,

the most powerful obstructions to the existence of such metrics are based on the index theory

for the spin-Dirac operator. Indeed, the Bochner-Lichnerowicz formula [27] implies that, on

a closed spin manifold N with positive scalar curvature, the spin-Dirac operator is invertible

and hence its index must vanish.

Rosenberg ([29], [30], [31]) refined this obstruction by using Dirac operators twisted with

flat Hilbert C∗-module bundles of finite type. Let π1 be the fundamental group of N and

let C∗R(π1) be the real maximal group C∗-algebra of π1. By twisting the spin-Dirac operator

on N with the canonical flat Hilbert C∗R(π1)-bundle over N , one obtains the Rosenberg

index obstruction αR(N) ∈ KO∗(C∗R(π1)). It was conjectured that this obstruction gives a

complete characterization of the existence of metrics of positive scalar curvature on closed

spin manifolds.

Conjecture 1.0.1 (Gromov-Lawson-Rosenberg). Let N be a closed spin manifold of di-

mension at least 5. Then N admits a metric of positive scalar curvature if and only if

αR(N) = 0.

In the celebrated work [39], Stolz proved this conjecture when N is simply connected.

Since then, many other cases have been proved. On the other hand, the conjecture is

not always true by the counterexample found by Schick in dimension 5 (cf. [34]). For a

1

comprehensive discussion of this topic, we refer to the survey papers [32] and [40].

The study of complete metrics of positive scalar curvature on a noncompact manifold M

is more complicated. In the case when M is a cylinder with compact base, Rosenberg and

Stoltz proposed the following conjecture.

Conjecture 1.0.2. ([33, Conjecture 7.1]) Let N be a closed manifold. If N × R admits

a complete metric of positive scalar curvature, then N admits a metric of positive scalar

curvature.

When N is enlargeable, this conjecture holds by results of Gromov and Lawson (see [19,

Corollary 6.13 and Theorem 7.5]). In [33, Section 7], Stolz and Rosenberg proved many other

cases by using the minimal surface technique. When the manifold N has a spin structure,

it is possible to use Dirac obstructions on N to construct obstructions to the existence of a

complete metric of positive scalar curvature on N ×R. In particular, the following theorem

holds.

Theorem 1.0.1. (Rosenberg, [32, Theorem 3.4]) Let N be a closed spin manifold. If N ×R

admits a complete metric of uniformly positive scalar curvature, then αR(N) = 0.

Notice that this theorem is a first step toward connecting Conjecture 1.0.1 with Conjec-

ture 1.0.2. Suppose, in fact, that N is a closed spin manifold satisfying Conjecture 1.0.1. If

N ×R admits a complete metric of uniformly positive scalar curvature, then Theorem 1.0.1

implies that N admits a metric of positive scalar curvature. A natural question to ask is

whether Theorem 1.0.1 holds under the weaker hypothesis that N × R admits a complete

metric of (not necessarily uniformly) positive scalar curvature.

In this dissertation, we work with complex C∗-algebras instead of real ones. Let C∗C(π1)

be the maximal complex group C∗-algebra associated to the fundamental group π1 of N . By

twisting the spin-Dirac operator on N with the canonical flat Hilbert C∗C(π1)-bundle of finite

type over N , we obtain an index obstruction αC(N) ∈ K0(C∗C(π1)). The first main result is

the following theorem.

Theorem A. Let N be a closed spin manifold. If N×R admits a complete metric of positive

scalar curvature, then αC(N) = 0.

2

Remark 1.0.2. The “real” version of this theorem, i.e. with αC(N) replaced by αR(N), would

allow us to deduce Conjecture 1.0.2 for all closed spin manifolds N verifying Conjecture 1.0.1.

The author plans to treat the case of real C∗-algebras in a future paper.

Remark 1.0.3. In [36], Schick and Zadeh used coarse index theory to study obstructions to

uniformly positive scalar curvature metrics on M in terms of suitable closed submanifolds

of arbitrary codimension. In particular, their approach allows one to deduce Theorem A for

complete metrics on M with uniformly positive scalar curvature. From this point of view,

the main novelty of this dissertation consists in extending this result to metrics whose scalar

curvature is positive and decays at infinity.

We deduce Theorem A from an abstract index theorem for Callias-type operators twisted

with Hilbert C∗-bundles of finite type. A Callias-type operator on a complete Riemannian

manifold M is an operator of the form P = D + Φ, where D is a Dirac operator and Φ is a

potential, such that P 2 is an operator of Schrödinger-type with potential uniformly positive

at infinity. This means that P 2 = D2 + V , where V is a bundle map uniformly positive

outside of a compact set. This condition implies that the spectrum of P is discrete near zero

so that P is Fredholm.

The study of such operators was initiated by Callias, [13], and further developed by many

authors, cf., for example, [6], [11], [2], [12]. Several generalizations and new applications of

the Callias-type index theorem were obtained recently, cf. [22], [14], [42], [23], [10]. For this

dissertation, the relevant property of Callias-type operators is that the computation of their

index can be reduced to the computation of the index of a Dirac-type operator on a suitable

codimension-one closed submanifold. In particular, the Callias-type index theorem, [2], [12],

states that the index of P is equal to the index of a certain Dirac operator DN induced by

the restriction of P to a suitable closed hypersurface N .

In this dissertation we suppose that A is a complex C∗-algebra, V is a Hilbert A-bundle

of finite type over M and PV is the operator obtained by twisting P with the bundle V . We

extend to this setting the theory of Callias-type operators.

Theorem B (Callias-type operators in C∗-algebras). We have:

(a) (Fredholmness) The operator PV is A-Fredholm. In particular, it has a well-defined

3

index class indA PV ∈ K0(A).

(b) (Callias-type theorem) Suppose that M is orientable and there is a partition M =

M− ∪N M+, where N = M− ∩M+ is a closed codimension-one submanifold of M and

M− is a compact submanifold, whose interior contains an essential support of Φ (see

Definition 2.3.1). If A is separable and σ-unital, then indA PV = indADN,i∗V in K0(A),

where i : N ↪→ M is the inclusion map and DN,i∗V is the Dirac operator DN twisted

with the pull-back bundle i∗V .

Remark 1.0.4. Suppose N is as in the hypothesis of Theorem A. Then the cylinder N × R

is endowed with a spin structure. We can use this structure to construct a Callias-type

operator P on N ×R. Moreover, suppose that V is the canonical Hilbert C∗(π1)-bundle on

N × R, where π1 is the fundamental group of N . Then by using a Bochner method it is

possible to show that the index class of PV vanishes and Theorem A follows from part (b)

of Theorem B (see Section 5.2).

Remark 1.0.5. Suppose A is a von Neumann algebra endowed with a finite trace τ . In [8],

it is shown that it is possible to use the trace τ to define a numerical index indτ PV ∈ R.

It would be interesting to understand if indτ PV descends from the K-theoretical index

indA PV ∈ K0(A) used in this dissertation.

Remark 1.0.6. In the setting of Remark 1.0.5, a Callias-type theorem has been proved in [7].

This result cannot be deduced from part (b) of Theorem B even in the case when the τ -index

of PV descends from the K-theoretical one since, in general, von Neumann algebras are not

separable.

The possibility of extending the analysis of Callias-type operators to Hilbert C∗-bundle

was left as an open question by Bunke in [12]. In particular, Bunke pointed out some issues

in proving both Fredholmness and the Callias-type theorem. For Fredholmness, the main

problem is to show the invertibility of P 2V + 1. To solve this issue, we make use of a result

of Hanke, Pape and Schick (cf. [20]) that guarantees that the operator P 2V + 1 has dense

range. In order to prove invertibility, we also show that the operator P 2V + 1 has a unique

self-adjoint extension.

4

When A = C, the proof of the Callias-type theorem used in both [2] and [12] consists of

two steps. In the first step, a “cut-and-paste” argument is used to reduce the initial problem

to a problem on the cylinder N×R. In the second step, a “separation of variables” argument

is used to show that the kernel (resp. cokernel) of the operator on the cylinder is isomorphic

to the kernel (resp. cokernel) of the operator on the base space. By using the K-theoretic

relative index theorem of Bunke [12], the “cut-and-paste” argument can be adapted to the

case when A is an arbitrary C∗-algebra.

The second problem pointed out by Bunke arises in the computations on the cylinder. In

the case of arbitrary C∗-algebras, in order to define the index classes of the operators on the

cylinder and on the base space we need to perturb these operators. This fact doesn’t allow

us to separate the variables. We change our point of view here. We formulate the index

classes of the operators on the cylinder and on the base space in a KK-theoretical setting

and make heavy use of the properties of the Kasparov intersection product to reduce the

twisted case to the untwisted one. In order to do these computations, we use the notion of

unbounded connection developed by Kucerovsky in [24]. To this end, we show that, under

suitable growth conditions of the endomorphism Φ, the operator PV defines an unbounded

Kasparov cycle.

The thesis is organized as follows. In Chapter 2, we show that the operator P 2V + 1 has a

unique self-adjoint extension. We use this fact to construct the class indA PV ∈ K0(A) (this

corresponds to point (a) of Theorem B). In Chapter 3 we show that the computation of this

class can be reduced to the computation of the index class of a model Callias-type operator.

This is the first step in proving part (b) of Theorem B. Chapter 4 is devoted to solving a

model problem on a cylinder. This concludes the proof of Theorem B. Finally, in Chapter 5

we prove a vanishing theorem from which we deduce Theorem A.

5

Chapter 2

The index of twisted Callias-type

operators

A Dirac-type operator on a noncompact manifold M is of Callias-type if its square is a

Schrödinger-type operator with a potential that is uniformly positive outside of a compact

set. In this chapter we start the study of operators of Callias-type twisted with Hilbert

A-bundles of finite type, where A is a C∗-algebra. To this end we study Schrödinger-type

operators twisted with a Hilbert A-bundle and prove that, under suitable growth conditions

of the potential, such operators have a unique self-adjoint extension. We use this fact to

define the index class of a twisted Callias-type operator PV . Our construction relies on an

approach due to Bunke. The main difficulty of the analysis done in this chapter is that

we work with unbounded operators on Hilbert A-modules and not on Hilbert spaces. This

means for instance that the graph of a closed operator does not need to be complementable.

This fact makes the analysis of self-adjoint extensions of differential operators much more

delicate.

The chapter is organized as follws. In Section 2.1 we review some background material.

In particular, we recall the notions of Hilbert A-module, Hilbert A-bundle and of connections

on such bundles. Finally, we review the construction of Dirac-type operators twisted with

a Hilbert A-bundle. In Section 2.2 we study twisted Schrödinger-type operators and show

that they have a unique self-adjoint extension. In Section 2.3 we define twisted Callias-type

operators and show that they are invertible at infinity in the sense of Bunke.

6

2.1 Twisted Dirac-type operators

This subsection is devoted to reviewing the construction of Dirac-type operators twisted with

a Hilbert A-bundle.

2.1.1 Hilbert A-modules

Let A be a C∗-algebra. In this section we recall the notion of Hilbert A-module and some

of its properties.

Definition 2.1.1. A pre-Hilbert A-module is a right A-module H equipped with an A-valued

inner product 〈·, ·〉 : H ×H → A satisfying the following properties:

(1) 〈x, ya〉 = 〈x, y〉a , x, y ∈ H , a ∈ A .

(2) 〈x1 + x2, y〉 = 〈x1, y〉 + 〈x2, y〉 , x1, x2, y ∈ H .

(3) 〈y, x〉 = 〈x, y〉∗ , x, y ∈ H .

(4) 〈x, x〉 ≥ 0 and 〈x, x〉 = 0⇔ x = 0 x ∈ H .

We say that H is a Hilbert A-module if it is a Banach space with respect to the norm

x 7→ |〈x, x〉|1/2A , x ∈ H . (2.1)

Example 2.1.2. Let A be a C∗-algebra.

(a) A is itself a Hilbert A-module with respect to the inner product 〈a, b〉 = a∗b, for a,

b ∈ A.

(b) The direct sum⊕

i∈I A is a pre-Hilbert A-module with respect to the inner product

〈(ai), (bi)〉 =∑i∈I

a∗i bi , (ai) , (bi) ∈⊕i∈I

A . (2.2)

(c) Let I = N in the previous example. The Hilbert A-module obtained as the completion

of⊕

i∈NA with respect to the norm

|(ai)| :=

∣∣∣∣∣n∑i=1

a∗i ai

∣∣∣∣∣ , (ai) ∈⊕i∈N

A

7

is called the standard countably generated Hilbert A-module and is denoted by HA. Its

elements are all sequences (ai) such that∑∞

i=0 a∗i ai converges.

(d) When, in example (b), I is a finite set, the inner product (2.2) defines a Hilbert A-

modules structure on An.

Definition 2.1.3. Given two Hilbert A-modules H1 and H2, we denote by LA(H1, H2) the

vector space of all operators T : H1 → H2 for which there is an adjoint operators T ∗ : H2 →

H1, with

〈Tx, y〉F = 〈x, T∗y〉 , x ∈ H1 , y ∈ H2 .

For a Hilbert A-module H, we also set LA(H) := LA(H,H).

Remark 2.1.4. (a) When A = C, the notion of Hilbert A-module coincide with the notion

of Hilbert space.

(b) Adjointable operators are automatically module homomorphisms.

(c) Every adjointable operator has closed graph, hence it is bounded.

(d) It is not true that every bounded operator between Hilbert A-modules is adjointable

(see example 2.1.5). This is a remarkable point where the theory of Hilbert A-modules

differs from the theory of Hilbert spaces.

(e) LA(H1, H2) is a Banach space and LA(H) is a C∗-algebra.

Example 2.1.5. In this example, taken from [28, pag. 447], we construct a bounded non-

adjointable operator between Hilbert A-modules. Let A = C([0, 1]), J := C0 ( ]0, 1[ ). The

direct sum H = A⊕ J is a Hilbert A-module with respect to the inner product

〈a1 ⊕ j1, a2 ⊕ j2〉 := a∗1a2 + j∗1j2 , ai ∈ A , ji ∈ J .

Consider the bounded operator T : H → H defined by T (a⊕ j) := j ⊕ 0.

Suppose that T admits an adjoint operator T ∗. In particular, T ∗(1⊕0) = x⊕y, for some

x ∈ A, y ∈ J . Then, for an arbitrary element a⊕ j ∈ H, we must have

j = 〈1⊕ 0, T (a⊕ j)〉 = 〈T ∗(1⊕ 0), a⊕ j〉 = 〈x⊕ y, a⊕ j〉 = x∗a+ y∗j .

Hence, x = 0 and j = 1, which is impossible since 1 doesn’t belong to J .

8

Definition 2.1.6. We say that a Hilbert A-module H is finitely generated projective if it

isomorphic as a Hilbert A-module to a closed orthogonal direct summand of An for suitable

n ∈ Z+.

Remark 2.1.7. In general, not all closed submodules of a Hilbert A-modules are comple-

mentable (see Example 2.1.8). This is a second important point where the theory of Hilbert

A-modules differs from the theory of Hilbert spaces.

Example 2.1.8. Consider the C∗ algebra A = C([0, 1]) as Hilbert module over itself. Then

the closed submodule J := C0 ( ]0, 1[ ) is not complementable, since J⊥ = {0}.

Now we define an important subclass of LA(H1, H2). We start with selecting some special

operators of rank one from H1 to H2. For x ∈ H2 and y ∈ H1, define the operator θx,y :

H1 → H2 through the formula

θx,y(z) := x 〈y, z〉H1 , z ∈ H1 .

We have θx,y ∈ LA(H1, H2), with θ∗x,y = θy,x. We denote by Θ(H1, H2) the linear span of all

operators H1 → H2 of the form θx,y.

Definition 2.1.9. The closure of Θ(H1, H2) in LA(H1, H2) is denoted by KA(H1, H2). Its

elements are called compact operators from H1 to H2. We also set KA(H) := KA(H,H) for

a Hilbert A-module H.

Example 2.1.10. We have:

(a) KA(A) ∼= A.

(b) KA(HA) ∼= K ⊗ A, where K denotes the compact operators in the Hilbert space l2.

2.1.2 Hilbert A-bundles and connections

In this subsection we recall the notion of Hilbert A-bundle and of connection on such bundles.

Definition 2.1.11. Let X be a locally compact Hausdorff space. A Hilbert A-bundle of

finite type V over X is a locally-trivial bundle π : V → X where the typical fiber is a

9

finitely generated projective Hilbert A-module H and there is an atlas of local trivializations

φj : V |Uj → Uj × E that are Hilbert A-module isomorphisms fiberwise.

When X is smooth, we also assume that the bundle V has a smooth structure, i.e. the

transition maps φi ◦ φ−1j : Ui ∩ Uj → LA(H) are smooth.

Definition 2.1.12. Let M be a smooth manifold and E be a finitely generated projective

Hilbert A-module. Consider the trivial Hilbert A-bundle of finite type M × E → M . For a

smooth section f ∈ C∞(M × E), the section

df ∈ C∞ (T ∗M ⊗ (M × E))

is defined in local coordinates by the formula

df :=∑i

dxi ⊗∂f

∂xi,

where (dxi) is a local orthonormal frame of T∗M .

Definition 2.1.13. A connection ∇ on a Hilbert A-bundle of finite type V → M is an

A-linear map

∇ : C∞(V ) −→ C∞(T ∗M ⊗ V )

which is a derivation with respect to multiplication with smooth sections of the trivial bundle

M × A, i.e.

∇(sf) = sdf + ∇(s)f , s ∈ C∞(V ) , f ∈ C∞(M ;A) .

We say that ∇ is a metric connection if

d 〈s1, s2〉 = 〈∇s1, s2〉 + 〈s1,∇s2〉 , s1, s2 ∈ C∞(V ) .

Example 2.1.14. The differential d defined in Definition 2.1.12 is a connection, called the

trivial connection on the trivial bundle M ×H. The trivial connection is actually a metric

connection with respect to the pointwise A-valued inner product

〈s1, s2〉 (x) = 〈s1(x), s2(x)〉H , s1 , s2 ∈ C∞(M ;A) .

10

2.1.3 Dirac-type operators twisted with Hilbert bundles

In this subsection we review the construction of Dirac-type operators twisted with Hilbert

A-bundles of finite type.

Let M be a complete Riemannian manifold and let E be a complex Dirac bundle over

M . This means that E is a complex hermitian vector bundle endowed with a Clifford action

c : T ∗M → End(E) of the cotangent bundle and a metric connection ∇E compatible with

the inner product of the fibers and satisfying the Leibniz rule (see [26, Definition 5.2.]). The

Dirac operator associated to this bundle is the formally self-adjoint operator /D ∈ Diff1(M ;E)

given by the composition

C∞c (M ;E)∇E−−→ C∞c (M ;T ∗M ⊗ E)

c−→ C∞c (M ;E).

Fix a potential Ψ ∈ C∞(M ; End(E)) and consider the Dirac-type operator

B := /D + Ψ. (2.3)

We also assume that the operator B is formally self-adjoint. Notice that this assumption

forces Ψ(x) to be a self-adjoint endomorphism of Ex for all x ∈M .

Let V →M be a Hilbert A-bundle of finite type endowed with a metric connection ∇V .

The product E ⊗ V is a Hilbert A-bundle of finite type. The Dirac operator /D twisted with

the bundle V is the operator /DV ∈ Diff1(M ;E ⊗ V ) defined through the composition

C∞c (M ;E)∇E⊗1+1⊗∇V−−−−−−−−→ C∞c (M ;T ∗M ⊗ E ⊗ V )

c⊗1−−−→ C∞c (M ;E). (2.4)

We also extend the potential Ψ to a section ΨV ∈ C∞(M ; EndA(E ⊗ V )) by setting

ΨV := Ψ⊗ 1.

Definition 2.1.15. The Dirac-type operator B = /D + Ψ twisted with the bundle V is the

operator BV := /DV + ΨV .

2.2 Schrödinger-type operators

In this section, we study operators of Schrödinger-type acting on Hilbert A-bundles of fi-

nite type. In Subsection 2.2.1 we define the notion of regular essential self-adjointness for

11

unbounded operators acting on Hilbert A-modules and prove that this notion ensure that

such operators have a unique self-adjoint extension. In Subsections 2.2.2, 2.2.3, and 2.2.4 we

show that, under suitable conditions on the potential, Schrödinger-type operators acting on

Hilbert A-bundles are regularly essentially self-adjoint.

2.2.1 Unbounded operators on Hilbert A-modules

Let H be a Hilbert A-module. An operator on H is an A-linear map T : Dom(T ) → H,

where Dom(T ) is an A-submodule of H called the domain of T . We say that T is densely

defined if Dom(T ) = H. If S and T are operators on H such that Dom(T ) ⊂ Dom(S) and

Sx = Tx for every x ∈ Dom(T ), we say that S is an extension of T .

An operator T on H is said to be closed if its graph is a closed submodule of H ⊕ H.

We say that T is closable if it admits a closed extension. In this case, the minimal closed

extension is called the closure of T and is denoted by T . Notice that

Dom(T)

={x ∈ H

∣∣∃(xn) ⊂ Dom(T ) with xn → x and Txn → Tx} . (2.5)For a densely defined operator T in H, we define the A-module

Dom (T ∗) ={y ∈ H

∣∣∃z ∈ H with 〈Tx, y〉 = 〈x, z〉 , ∀x ∈ Dom(T )} . (2.6)The element z on the right hand side is unique and we define the operator D∗ : Dom(D∗)→

H by setting T ∗y = z. The operator T ∗ is called the adjoint of T . It is closed, but in general

not densely defined.

The main difference from the case when H is a Hilbert space is that even when T is closed

its graph does not need to be complementable. This fact justifies the following definition.

Definition 2.2.1. We say that a closed densely defined operator T : Dom(T ) ⊆ H → H is

regular if T ∗ is densely defined and the graph of T is complementable in H ⊕H.

A densely defined operator T is said to be symmetric if T ⊆ T ∗ and self-adjoint if

T = T ∗. Symmetric operators are closable and self-adjoint operators are closed. In the next

proposition we recall some properties of regular operators: for the proof, see [15, Section 9].

Proposition 2.2.2 (Properties of regular operators). We have:

12

(a) if both T and T ∗ are densely defined, then T is regular if and only if T ∗ is regular;

(b) if T is closed, densely defined and symmetric, then T is regular if and only if the

operators T ± i have dense range;

(c) if T is regular, then T ∗∗ = T ;

(d) if T is regular, then T ∗T is regular and self-adjoint;

(e) if T is regular, then T ∗T + µ2 : Dom(T ∗T ) → H is surjective for every real number

µ 6= 0.

Definition 2.2.3. We say that a densely defined symmetric operator T on H is regularly

essentially self-adjoint if T ∗ is regular and symmetric.

The next proposition tells us that Definition 2.2.3 catches the right notion of essential

self-adjointness for operators acting on Hilbert A-modules.

Proposition 2.2.4. Let T be a densely defined symmetric operator on H. Then the following

are equivalent:

(i) T is regularly essentially self-adjoint;

(ii) the minimal closure T is a regular, self-adjoint operator.

Moreover, if T is regularly essentially self-adjoint, then T is the only self-adjoint extension

of T .

Proof. The implication (ii)⇒ (i) is trivial, for T ∗ = T ∗. Let us prove that (i)⇒ (ii). Since

T is symmetric and densely defined, both T and T ∗ are densely defined. Since T ∗ = T∗

and

T ∗ is regular, T is regular by part (a) of Proposition 2.2.2. Since T is symmetric, then T is

also symmetric, i.e. T ⊆ T ∗. Since T ∗ = T ∗ is symmetric by hypothesis, we have

T ⊆ T ∗ ⊆ T ∗∗ = T ,

where the last equality follows from part (c) of Proposition 2.2.2. Therefore, T is self-adjoint.

13

The proof of the second part of the proposition is analogous to the case when A = C.

Let R be a self-adjoint operator extending T . Since, by the first part of the proposition, T ∗

is symmetric, we have

T ⊆ R = R∗ ⊆ T ∗ = T .

Now the thesis follows from the fact that T is the minimal closure of T .

2.2.2 Operators of Schrödinger-type

Let M , E, V be as in Subsection 2.1.3. Suppose BV is a twisted Dirac-type operator. Fix a

real number µ 6= 0 and consider the Schrödinger-type operator

Hµ := B2V + µ

2 . (2.7)

Notice that C∞c (M ;E ⊗ V ) has the structure of a pre-Hilbert A-module with respect to the

A-valued inner product

〈u, v〉0 :=∫M

〈u(x), v(x)〉x dµ(x) , u, v ∈ C∞c (M ;E ⊗ V ) ,

where dµ(x) is the smooth measure induced by the Riemannian metric on M and 〈·, ·〉xdenotes the A-valued inner product of the fiber Ex ⊗ Vx. We denote by H0 the Hilbert

A-module obtained as the completion of C∞c (M ;E ⊗ V ) with respect to the norm

‖u‖0 :=√| 〈u, u〉0 |A , u ∈ C

∞c (M ;E ⊗ V ) ,

where | · |A denotes the norm of the C∗-algebra A. We view HF as an unbounded operator

on H0 with initial domain C∞c (M ;E ⊗ V ). When A = C, H0 is a Hilbert space and it is

a classical fact that HF has a unique extension to a self-adjoint operator on H0 (cf. for

instance [9]). The next theorem extends this result to the case when A is an arbitrary

C∗-algebra.

Theorem 2.2.5. The operator Hµ is regularly essentially self-adjoint.

Remark 2.2.6. In this dissertation, we focus on the case of Schrödinger-type operators with

uniformly bounded potential. When A = C, the essential self-adjointness of Schrödinger-

type operators holds for a larger class of potentials (see [9] for an overview on the topic).

The question whether the same thing holds when A is an arbitrary C∗-algebra is left open.

14

2.2.3 Regularity of H∗µ

In this subsection we show that the closed operator HF is regular. To this end, we need

some information about the operator BV . We regard BV as an unbounded operator on H0

with initial domain C∞c (M ;E ⊗ V ). In the case when A = C, it is a classical fact that BVhas a unique self-adjoint extension (see [19]). The next theorem extends this result to the

case of arbitrary complex unital C∗-algebras.

Theorem 2.2.7 (Hanke-Pape-Schick, [20]). The closure BV of the operator BV is a regular

self-adjoint operator. Moreover, it is the only self-adjoint extension of BV .

Remark 2.2.8. This theorem has been recently generalized by J. Ebert to a larger class of

elliptic first order differential operators acting on Hilbert A-bundles of finite type (see [17,

Theorem 1.14]).

Proposition 2.2.9. The operator H∗µ is regular.

Proof. By part (b) of Proposition 2.2.2, to prove the thesis we need to show that the operators

H∗µ ± i : Dom(H∗µ)→ H0 have dense range.

To this end, we compare H∗µ with the operator BV ◦ BV + µ2. By Theorem 2.2.7 and

part (d) Proposition 2.2.2 the operator BV ◦BV +µ2 is regular and self-adjoint. By part (b)

of Proposition 2.2.2, it follows that the operators

Dom(BV ◦BV + µ2

) BV ◦BV +µ2± i−−−−−−−−−−→ H0 (2.8)have dense range. Moreover,

Hµ ⊆ BV ◦BV + µ2 =(BV ◦BV + µ2

)∗ ⊆ H∗µ .This means that Dom

(BV ◦BV + µ2

)is a submodule of Dom(H∗F ) and that the operators

H∗F and BV ◦BV +µ2 coincide on Dom(BV ◦BV + µ2

). Since the operators (2.8) have dense

range, the operators

Dom(H∗µ)H∗µ± i−−−−−−→ H0

have dense range as well, from which the thesis follows.

15

2.2.4 Symmetricity of H∗µ

In this subsection we show that the operator H∗µ is symmetric and conclude the proof of

Theorem 2.2.5. To this end, we need some information about the asymptotic behavior of

sections in Dom(H∗µ).

For k ∈ Z+, let Hkloc denote the A-module consisting of measurable sections u of E ⊗ V

such that ψ u is in Hk for all ψ ∈ C∞c (M). In [18], it is shown that

Dom(H∗µ)⊆ H2loc . (2.9)

In particular, BV s ∈ H0loc for s ∈ Dom(H∗µ).

In the next lemmas we study the decay of BV s at infinity. Let us first introduce some

notation. Let σ(BV ) be the leading symbol of the operator BV . For ξ ∈ T ∗xM , it defines an

operator σ(BV )(ξ) ∈ LA(Ex) (see [38, Section 2.1]). Given φ ∈ C∞c (M), define the A-linear

operator

σ(BV )(dψ) : H0 −→ H0

by setting (σ(BV )(dψ)s

)(x) := σ(BV )(x, dψ(x))s(x) (2.10)

for s ∈ C∞c (M ;E ⊗ V ) and extending by continuity.

Lemma 2.2.10. Let ψ ∈ C∞c (M). Then σ(BV )(dψ) ∈ LA(H0) and

‖σ(BV )(dψ)‖B(H0) ≤ ‖dψ‖∞ . (2.11)

Proof. Notice that

σ(BV )(x, ξ) = i c(ξ)⊗ idV , (x, ξ) ∈ T ∗M , (2.12)

where c(ξ) is Clifford multiplication by ξ. Hence, σ(BV )(dψ) is adjointable. Moreover,

‖σ(BV )(dψ) s‖20 =∣∣∣∣∫M

‖dψ(x)‖2g 〈s(x), s(x)〉x dµ(x)∣∣∣∣A

≤ ‖dψ‖2∞ ‖s‖20 ,

from which the thesis follows.

Since the Riemannian metric on M is complete, by [37, Proposition 4.1] there exists a

sequence {φk}∞k=0 of compactly supported real-valued smooth functions on M such that

16

(C.1) 0 ≤ φk(x) ≤ 1, for all k ∈ Z+ and all x ∈M ;

(C.2) there exists a sequence {Lk}∞k=0 of compact sets exhausting M such that φk = 1 on LKand supp(φk) ⊆ Lk+1;

(C.3) the sequence {dφk}∞k=0 converges to 0 in the ‖ · ‖∞-norm.

Notice that, for s ∈ Dom(H∗µ), the section φk BV s ∈ H0 is well-defined by (2.9).

Lemma 2.2.11. Let s ∈ Dom(H∗µ)

and let {φk}∞k=0 be a sequence of compactly supported

smooth functions satisfying (C.1) and (C.3). Then there exists a constant C such that

‖φkBV s‖0 ≤ C , ∀k ∈ Z+ . (2.13)

Proof. For k ∈ Z+, we have

〈φkBV s, φkBV s〉0 = 〈BV (φ2kBV s), s〉0

= 〈φ2k B2V s, s〉0 − 2i 〈σ(BV )(dφk)φkBV s, s〉0

= 〈φ2kHµs, s〉0 − 〈φ2k µ

2 s, s〉0 − 2i 〈σ(BV )(dφk)φkBV s, s〉0 ,

(2.14)

where σ(BV )(dφk) is the operator defined by (2.10). Taking the A-norm in (2.14) and using

Lemma 2.2.10 and (C.1), we deduce

‖φk BV s‖20 ≤∣∣〈φkHµs, φk s〉0∣∣A + ∣∣〈φ2k µ2 s, s〉0∣∣A + 2 ∣∣ 〈σ(BV )(dφk)φkBV s, s〉 ∣∣A

≤ ‖φkHµs‖0 ‖φk s‖0 + µ2 ‖s‖20 + 2 ‖σ(BV )(dφk)φkBV s‖0 ‖s‖0

≤(∥∥H∗µs∥∥0 + µ2 ‖s‖0) ‖s‖0 + 2 ‖dφk‖∞ ‖φkBV s‖0 ‖s‖0 .

(2.15)

Using the inequality ab ≤ 12a2 + 1

2b2, we get

2 ‖dφk‖∞ ‖φkBV s‖0 ‖s‖0 ≤1

2‖φkBV s‖20 + 2 ‖dφk‖2∞ ‖s‖20 . (2.16)

From (2.15) and (2.16), we obtain

‖φk BV s‖20 ≤ 2(∥∥H∗µs∥∥0 + µ2 ‖s‖0) ‖s‖0 + 4 ‖dφk‖2∞ ‖s‖20 .

By (C.3), the previous inequality implies the thesis.

Proposition 2.2.12. The operator H∗µ is symmetric.

17

Proof. Let s1, s2 ∈ Dom(H∗µ). We need to show that〈H∗µ s1, s2

〉0

=〈s1, H

∗µs2〉

0. (2.17)

Let {φk}∞k=0 be a sequence of compactly supported functions satisfying (C.1)–(C.3). From (2.9)

it follows that B2V s ∈ H0loc. Therefore, the inner products 〈φks1, B2V s2〉0, 〈φks1, Fs2〉0 are

well-defined and〈φks1, H

∗µs2〉

0=〈B2V (φks1), s2

〉0

+〈µ2 φk s1, s2

〉0

= 〈BV (φks1), BV s2〉0 +〈φks1, µ

2 s2〉

0

= 〈φkBV s1, BV s2〉0 − i 〈σ(BV )(dφk)s1, BV s2〉0 +〈φks1, µ

2 s2〉

0,

(2.18)

where σ(BV )(dφk) is the operator defined in (2.10). Similarly,〈H∗µs1, φks2

〉0

= 〈BV s1, φkBV s2〉0 − i 〈BV s1, σ(BV )(dφk)s2〉0 +〈µ2 s1, φks2

〉0. (2.19)

From (2.18) and (2.19), we deduce〈φks1, H

∗µs2〉

0−〈H∗µs1, φks2

〉0

= i 〈BV s1, σ(BV )(dφk)s2〉0 − i 〈σ(BV )(dφk)s1, BV s2〉0

= i 〈φk+1BV s1, σ(BV )(dφk)s2〉0 − i 〈σ(BV )(dφk)s1, φk+1BV s2〉0 ,

where the second equality holds since, by (C.2), φk+1 = 1 on the support of φk. Taking the

A-norm in the previous equality and using Lemma 2.2.10, we obtain∣∣∣ 〈φks1, H∗µs2〉0 − 〈H∗µs1, φks2〉0 ∣∣∣A ≤≤ ‖φk+1BV s1‖0 ‖σ(BV )(dφk)s2‖0 + ‖σ(BV )(dφk)s1‖0 ‖φk+1 BV s2‖0 ≤ C ‖dφk‖∞ ,

for a suitable constant C. Hence,〈φks1, H

∗µs2〉


〉0−−−→ 0 , as k →∞ . (2.20)

From (C.1) and (C.2), it follows that φksj → sj in H0-norm. Therefore,〈φks1, H

∗µs2〉


〉0−−−→

〈s1, H

∗µs2〉

0−〈H∗µs1, s2

〉0, as k →∞ . (2.21)

Finally, (2.20) and (2.21) imply (2.17).

Proof of Theorem 2.2.5

It follows from Propositions 2.2.12 and 2.2.9.

18

2.3 Twisted Callias-type operators

We define a particular class of twisted Dirac-type operators, called Callias-type. The study

of these operators is the main object of this thesis. This section is devoted to define the

index class of such operators.

2.3.1 Twisted operators of Callias-type

Let M be an odd-dimensional complete Riemannian manifold and let Σ be an ungraded

complex Dirac bundle over M . Let D ∈ Diff1(M ; Σ) be a formally self-adjoint Dirac-type

operator (see Subsection 2.1). Fix a self-adjoint potential Φ ∈ C∞(M ; End(Σ)) and consider

the operator

P :=

0 D − iΦD + iΦ 0

. (2.22)Notice that P ∈ Diff1(M ; Σ ⊕ Σ) is of Dirac-type and formally self-adjoint. Let V be as in

Subsection 2.1 and consider the twisted Dirac-type operator

PV :=

0 DV − iΦVDV + iΦV 0

, (2.23)where ΦV := Φ⊗ idV .

We are now ready to define the main object of this dissertation.

Definition 2.3.1. The endomorphism Φ is said to be admissible for the pair (Σ, D) if

(i) the commutator [D,Φ] := DΦ− ΦD is an endomorphism of Σ;

(ii) there exist a constant d > 0 and a compact set K ⊂M such that

Φ2(x) ≥ d + ‖[D,Φ](x)‖ , x ∈M \K . (2.24)

In this case, we say that K is an essential support for Φ and that the operator PV de-

fined in (2.23) is the twisted Callias-type operator associated to the admissible quadruple

(Σ, D,Φ, V ). If we can choose K = ∅, we say that Φ has empty essential support.

19

2.3.2 Bounded Kasparov modules

In this subsection we quickly review the notion of bounded Kasparov module and KK-group.

Definition 2.3.2. Let A and B be graded C∗-algebras. A bounded Kasparov module for

(A,B) is a triple (E, φ, F ), where E is a countably generated graded Hilbert B-module, φ :

A → LB(E) is a graded ∗-morphism and F is an operator in LB(E) of degree one such

that [F, φ(a)], (F 2 − 1)φ(a) and (F − F ∗)φ(a) are in KB(E), for all a ∈ A. We denote by

KK(A,B) the set of all homotopy classes of bounded Kasparov modules for (A,B).

Remark 2.3.3. (a) The structure group of KK(A,B) is described in [4, Proposition 17.3.3].

(b) For the notion of homotopy between bounded Kasparov modules we refer to [4, Defi-

nition 17.3.1]. For this dissertation the properties are relevant:

(i) For t ∈ [0, 1], let (E, φ, Ft) be a family of Kasparov modules for (A,B). If the

family of operators {Ft} is norm-continous, then (E, φ, Ft) is a homotopy be-

tween (E, φ, F0) and (E, φ, F1). In particular, the class [E, φ, Ft] ∈ KK(A,B) is

independent of t;

(ii) If (E, φ, F ) is a bounded Kasparov module for (A,B) and K ∈ KB(E), then

(E, φ, F ) and (E, φ, F +K) are homotopy equivalent and define the same class in

KK(A,B).

The group KK(C, A) is isomorphic to K0(A) (see [4, Section 17.5.4]).

Example 2.3.4. (Mǐsčenko-Fomenko index) Let M , E, and V be as in Subsection 2.1.3.

Suppose that E is Z2-graded, i.e. E = E+ ⊕ E−. In this case, the twisted Dirac operator

/DV is odd and formally self-adjoint, i.e.

/DV =

0 /D−V/D

+V 0

,where /D

+V : C

∞(E+ ⊗ V ) → C∞(E− ⊗ V ) and /D−V is the formal adjoint of /D+V . Let

H0(E±⊗V ) be the Hilbert A-module of the square integrable sections of the bundle E±⊗V

(see Subsection 2.2.2). When M is a closed manifold, Mǐsčenko and Fomenko (see [18])

20

associated to this operator an index class indA /DV in KK(C, A) = K0(A) represented by

the bounded Kasparov module(H0(E+ ⊗ V )⊕H0(E− ⊗ V ), 1, /DV

(1 + /D

2V

)−1/2)for the pair of algebras (C, A). Here, 1 is scalar multiplication by complex numbers.

2.3.3 Sobolev spaces

Fix a nonnegative integer l. We use the operator BV to define the A-valued inner product

〈u, v〉l :=l∑

k=0

∫M

〈(BkV u

)(x),

(BkV v

)(x)〉xdµ(x), u, v ∈ C∞c (M ;E ⊗ V ) ,

where dµ(x) is the smooth measure induced by the Riemannian metric on M and 〈·, ·〉xdenotes the A-valued inner product of the fiber Ex ⊗ Vx. Endowed with this inner product,

C∞c (M ;E⊗V ) has a pre-Hilbert A-module structure. We denote by H l the Hilbert A-module

obtained as the completion of C∞c (M ;E ⊗ V ) with respect to the norm

‖u‖l :=√| 〈u, u〉l |A , u ∈ C

∞c (M ;E ⊗ V ) ,

where | · |A denotes the norm of the C∗-algebra A.

Given an operator T ∈ LA(H i, Hj), we denote by ‖T‖B(Hi,Hj) the operator norm of T as

a bounded operator H i → Hj. Finally, we set ‖ · ‖B(Hi) := ‖ · ‖B(Hi,Hi).

2.3.4 A-index of twisted Dirac-type operators: Bunke’s approach

Let q : M → R be a smooth function which is constant outside of a compact set and consider

the operator B2V + q. Notice that the Sobolev space H2 is isomorphic, as a Banach space,

to Dom(P 2V + q

)endowed with the graph norm. Therefore, the unique regular self-adjoint

extension of B2V +q, that we denote again by B2V +q, defines a bounded adjointable operator

B2V + q : H2 −→ H0 . (2.25)

Definition 2.3.5. We say that the operator B2V is invertible at infinity if there exists a func-

tion f ∈ C∞c (M) such that the operator B2V + f is invertible and(B2V + f

)−1 ∈ LA(H0, H2).21

In this case, Bunke associated to the operator BV a K-theoretical index class. The

construction of this class makes use of Kasparov’s KK-theory. For the notion of bounded

Kasparov module and KK-group, we refer to [4].

Theorem 2.3.6 (Bunke, [12]). Suppose B2V is invertible at infinity and let f ∈ C∞c (M) be

as in Definition 2.3.5. Then the triple(H0, 1, BV

(B2V + f

)−1/2)(2.26)

is a bounded Kasparov module for the pair of algebras (C, A). Moreover, the class in

KK(C, A) = K0(A) defined the triple (2.26) is independent of the choice of the function

f ∈ C∞c (M) such that the operator B2V + f is invertible and(B2V + f

)−1 ∈ LA(H0, H2). Inthis case, the A-index of BV is the K-theoretical class

indABV :=[H0, 1, BV

(B2V + f

)−1/2] ∈ KK(C, A) = K0(A) , (2.27)where f is any function in C∞c (M) such that the operator B

2V + f is invertible and

(B2V +

f)−1 ∈ LA(H0, H2).

2.3.5 Invertibility of B2V + f

In this subsection we give a sufficient condition for the invertibility of the operator B2V + f ,

where f : M → R is a uniformly bounded smooth function. In our analysis we identify H1

with Dom(PV)

and regard the unique regular self-adjoint extension of BV , that we denote

again by BV , as a bounded adjointable operator

BV : Hj+1 −→ Hj , (2.28)

for j = 0, 1.

Lemma 2.3.7. For every nonzero real number µ, the operator B2V + µ2 is invertible. More-

over, (B2V + µ2)−1 ∈ LA(H0) ∩ LA(H0, H2).

Proof. Fix an element v ∈ H2. Then both v and BV v are in H1. Since BV is self-adjoint,

we obtain 〈(B2V + µ

2)v, v〉

0= 〈BV v,BV v〉0 +

〈µ2v, v

〉0≥ µ2 〈v, v〉0 . (2.29)

22

Applying the A-norm to the previous inequality, we get

µ2‖v‖20 = µ2 | 〈v, v〉0 |A ≤∣∣〈(B2V + µ2) v, v〉0∣∣A ≤ ∥∥(B2V + µ2) v∥∥0 ‖v‖0 .

Hence, ∥∥(B2V + µ2) v∥∥0 ≥ µ2 ‖v‖0 , v ∈ H2 . (2.30)This inequality implies that the operator B2V + µ

2 is injective and a fortiori invertible on

its range. Moreover, by part (e) of Proposition 2.2.2 the operator B2V + µ2 is surjective, i.e.

its range coincides with the whole space H0. By the open mapping theorem, the A-linear

operator (B2V + µ

2)−1

: H0 −→ H2

is bounded. Since the inclusion H2 ↪→ H0 is bounded, (B2V + µ2)−1

is bounded also as an

operator H0 → H0. It remains to show adjointability.

Let us first show that (B2V + µ2)−1

is adjointable as operator H0 → H0. From (2.29), it

follows that〈u,(B2V + µ

2)−1

u〉

0=〈(B2V + µ

2) (B2V + µ

2)−1

u,(B2V + µ

2)−1

u〉

0

≥ µ2〈(B2V + µ

2)−1

u,(B2V + µ

2)−1

u〉

0≥ 0 ,

(2.31)

for every u ∈ H0. By [15, Lemma 4.1], Inequality (2.31) implies that (B2V + µ2)−1

is a

self-adjoint element in LA(H0), i.e.〈(B2V + µ

2)−1

u, v〉

0=〈u,(B2V + µ

2)−1

v〉

0u, v ∈ H0. (2.32)

Let us finally show that (B2V + µ2)−1

is adjointable as operator H0 → H2. Fix u ∈ H0,

w ∈ H2 and set Q := (B2V + µ2)−1

. Observe that BVw ∈ H1 and B2Vw ∈ H0. Since BV is

self-adjoint and using (2.32), we obtain

〈Qu,w〉2 =〈B2VQu,B

2Vw〉

0+ 〈BVQu,BVw〉0 + 〈Qu,w〉0

=〈(B2V + µ

2)Qu,B2Vw

〉0

+(1− µ2

) 〈Qu,B2Vw

〉0

+ 〈u,Qw〉0

=〈u,B2Vw

〉0

+(1− µ2

) 〈u,QB2Vw

〉0

+ 〈u,Qw〉0

=〈u,{B2V +

(1− µ2

)QB2V + Q

}w〉

0,

from which it follows that Q = (B2V + µ2)−1

is adjointable also as an operator H0 → H2.

23

Lemma 2.3.8. Let µ be a nonzero real number and let f : M → R be a uniformly bounded

smooth function such that µ2 > ‖f‖∞. Then the operator B2V +µ2+f is invertible. Moreover,

(B2V + µ2 + f)

−1 ∈ LA(H0) ∩ LA(H0, H2).

Proof. By Lemma 2.3.7, B2V + µ2 is invertible and the inverse is in both LA(H0) and

LA(H0, H2). In particular, we can write

B2V + µ2 + f =

{Id +f

(B2V + µ

2)−1}(

B2V + µ2). (2.33)

Since µ2 > ‖f‖∞, we have〈(B2V + µ

2)s, s〉

0≥ µ2 〈s, s〉0 > ‖f‖∞ 〈s, s〉0 , s ∈ H

2 .

By the Cauchy-Schwartz inequality for Hilbert A-modules, we obtain∥∥(B2V + µ2)−1∥∥B(H0) < ‖f‖∞ . (2.34)Since (B2V + µ

2)−1 ∈ LA(H0) and f ∈ L∞(M), then also f (B2V + µ2)

−1 ∈ LA(H0) and by

(2.34) we deduce ∥∥∥ f (B2V + µ2)−1∥∥∥B(H0) < 1 .Therefore, the operator Id +f (B2V + µ

2)−1

is invertible with bounded inverse given by the

Neumann series∞∑k=0

(−1)k{f(B2V + µ

2)−1}k

.

Since this series converges in norm and each summand is adjointable, it defines an element

in LA(H0). Since(B2V +µ

2)−1

is in LA(H0)∩LA(H0, H2), from (2.33) we finally deduce that

B2V +µ2 +f is invertible with inverse in LA(H0)∩LA(H0, H2) given by the norm convergent

series (B2V + µ

2)−1 ∞∑

k=0

(−1)k{f(B2V + µ

2)−1}k

.

Theorem 2.3.9. Let f : M → R be a uniformly bounded smooth function such that there

exists and a constant c > 0 satisfying〈(B2V + f

)s, s〉

0≥ c 〈s, s〉0 , s ∈ H

2 . (2.35)

Then B2V + f is invertible with(B2V + f

)−1 ∈ LA(H0) ∩ LA(H0, H2).24

Proof. Pick µ 6= 0 such that µ2 > ‖f‖∞. By Lemma 2.3.8, the operator B2V + µ2 + f is

invertible and (B2V + µ2 + f)

−1 ∈ LA(H0) ∩ LA(H0, H2). In particular, we can write

B2V + f =[Id −µ2

(B2V + µ

2 + f)−1] (

B2V + µ2 + f

). (2.36)

By Condition (2.35), we have

〈(B2V + µ

2 + f)s, s〉

0≥(µ2 + c

)〈s, s〉0 .

By the Cauchy-Schwartz inequality for Hilbert A-modules, we obtain∥∥∥µ2 (B2V + µ2 + f)−1∥∥∥B(H0) ≤ µ2µ2 + c < 1 . (2.37)Using a Neumann series in the same fashion as in the proof of Lemma 2.3.8, from (2.36)

and (2.37) we deduce the thesis.

2.3.6 A-index of twisted Callias-type operators

In order to define the index of a Callias-type operator PV , we show that the operator P2V

is invertible at infinity. When A = C, in [12] it was shown that Conditions (i) and (ii)

of Definition 2.3.1 imply the invertibility at infinity of P 2V . In [12, page 258], the question

whether the same is true when A is an arbitrary C∗-algebra is left open. We give a positive

answer to this question and use Formula (2.27) to associate an index class to twisted Callias-

type operators. The next theorem corresponds to part (a) of Theorem B.

Theorem 2.3.10. Let M be a complete Riemannian metric and let PV be the Callias-

type operator associated to an admissible quadruple (Σ, D,Φ, V ). Then the operator P 2V is

invertible at infinity. Therefore, PV has an index class indA PV ∈ K0(A) defined by (2.27).

Proof. Set

I :=

1 00 1

J := −i 0

0 i

. (2.38)With this notation, we have

P 2V =(D2V + Φ

2V

)I +

([DV ,ΦV ]

)J . (2.39)

25

In [7] (see the proof of Lemma 8.2) it is shown that

[DV ,ΦV

]=[D,Φ

]⊗ idV . (2.40)

Since PV is of Callias-type, we can choose a compact set K ⊂ M and a constant c > 0

satisfying

Φ2(x)−∥∥[D,Φ](x)∥∥ ≥ c , x ∈M \K . (2.41)

Let f : M → [0,∞) be a compactly supported smooth function such that

f(x) −∥∥[D,Φ](x)∥∥ ≥ c , x ∈ K . (2.42)

By Equations (2.39) and (2.40) we get

〈(B2V + f

)s, s〉

0≥〈{(

Φ2 + f − ‖[D,Φ]‖)⊗ idV

}s, s〉

0. (2.43)

From Equation (3.10), we obtain

Φ2(x) + f(x)−∥∥[D,Φ](x)∥∥ ≥ Φ2(x)− ∥∥[D,Φ](x)∥∥ ≥ c , x ∈M \K . (2.44)

From Equation (3.11), we get

Φ2(x) + f(x)−∥∥[D,Φ](x)∥∥ ≥ f(x)− ∥∥[D,Φ](x)∥∥ ≥ c , x ∈ K . (2.45)

Finally, from Equations (2.43), (2.44) and (2.45) we deduce that Condition (2.35) is satisfied.

Now the thesis follows from Theorem 2.3.9.

26

Chapter 3

Reduction to the cylinder

The next two chapters are devoted to the computation of the index of a twisted operator

of Callias-type PV over a complete orientable manifold M . We assume there is a partition

M = M− ∪N M+, where N = M− ∩ M+ is a closed hypersurface and M− is a compact

submanifold with boundary whose interior contains an essential support of Φ. The Callias-

type index theorem states that the computation of the index class of PV can be reduced to

the computation of the A-index of a suitable elliptic differential operator on N . Chapters 3

and 4 are devoted to proving this theorem. In this chapter we reduce the computation of

indA PV to the computation of the A-index of a model operator, i.e. a twisted Callias type

operator MVN on the cylinder N × R. In Chapter 4 we define a twisted Dirac operator on

N and show that its A-index coincides with indA MVN .

The present chapter is organized as follows. In Section 3.1 we formulate the Callias-type

index theorem. In Section 3.2 we prove some stability properties of the A-index of twisted

Callias-type operators. In Section 3.3 we use these properties to show that the computation

of indA PV can be reduced to the computation of the A-index of a Callias-type operator on

the cylinder N × R.

3.1 A Callias-type theorem

Let (Σ, D,Φ, V ) be an admissible quadruple on a complete oriented Riemannian manifold

M and let PV be the associated Callias-type operator. Suppose that there is a partition

27

M = M− ∪N M+, where N = M− ∩M+ is a smooth compact hypersurface and M− is a

compact submanifold, whose interior contains an essential support of Φ. We want to use our

data to construct a twisted Dirac-type operator on N and use this operator to compute the

class indA PV .

Let ΣN be the restriction of Σ to N ⊂M . Condition (ii) of Definition 2.3.1 implies that

zero is not in the spectrum of Φ(x) for all x ∈ N . Therefore, we have the splitting

ΣN = ΣN+ ⊕ ΣN− , (3.1)

where ΣN+, ΣN− are respectively the positive and negative eigenbundles of Φ.

By Condition (i) of Definition 2.3.1, the endomorphism Φ commutes with the Clifford

multiplication. Hence, c(ξ) : ΣN± → ΣN± for all ξ ∈ T ∗M∣∣N

. It follows that both bundles,

ΣN+ and ΣN−, inherit the Clifford action of T∗M . In particular, Clifford multiplication

by the unit normal vector field pointing at the direction of M+ defines an endomorphism

γ : ΣN± → ΣN±. Since γ2 = −1, the endomorphism α := −iγ induces a grading

ΣN± = Σ+N± ⊕ Σ

−N± , (3.2)

where ΣN± is the span of the eigenvectors of α with eigenvalues ±1.

We use the Riemannian metric on M to identify T ∗N with a subspace of T ∗M . Then the

Clifford action of T ∗N on ΣN± is graded with respect to this grading, i.e. c(ξ) : Σ±N± → Σ

∓N±

for all ξ ∈ T ∗N . Let ∇ΣN be the connection on ΣN obtained by restricting the connection

on Σ. It does not, in general, preserve decomposition (3.1). We define a connection ∇ΣN±

on the bundle ΣN± by setting

∇ΣN±s± := prΣN±(∇ΣNs±

), s± ∈ C∞(N ; ΣN±) , (3.3)

where prΣN± is the projection onto the bundle T∗N ⊗ ΣN±. By [1, Lemma 2.7] (see also [7,

Section 5.1]), ΣN+ and ΣN− carry a Z2-graded Dirac bundle structure.

We denote by DN+, DN− the Dirac operators on N associated respectively with the

bundles ΣN+ and ΣN−. Notice that the operators DN± are odd with respect to the grading

(3.2), i.e. they have the form

DN± =

0 D−N±D+N± 0

,28

where D+N± (respectively D−N±) is the restriction of DN± to Σ

+N± (respectively Σ

−N±).

Let VN be the restriction of V to N . It is a Hilbert A-bundle of finite type endowed

with a connection ∇VN obtained by pulling back the connection ∇V . Consider the operator

DN+,VN obtained by twisting the Dirac operator DN+ with the bundle VN . In the classical

paper [18], Mǐsčenko and Fomenko showed that the operator DN+,VN is A-Fredholm and it

has a well-defined index class indADN+,VN ∈ K0(A) (see also [35, Section 5]).

The next theorem is the main result of this dissertation. In the case when A = C it was

proved in [2, Theorem 1.5] and [12, Theorem 2.9]. When A is a von Neumann algebra with

a finite trace, the result has been recently proved in [7].

Theorem 3.1.1 (Callias-type theorem in C∗-algebras). Suppose that the C∗-algebra A is

separable. Then

indA PV = indADN+,VN . (3.4)

Remark 3.1.2. This theorem corresponds to part (b) of Theorem B.

Remark 3.1.3. The proof of this theorem consists of two steps. In the first step, that we

present in Section 3.3, we reduce the computation of indA PV to the computation of the

A-index of a model operator MVN , which is a Callias-type operator on the cylinder N ×

R. The second step, that we present in Chapter 4, consists of solving a problem on the

cylinder, i.e. proving Theorem 3.1.1 for the operator MVN . Our computations on the cylinder

require reformulating the problem in a KK-theoretical setting and use the properties of the

intersection product. This is the reason why, in the hypothesis of Theorem 3.1.1, we require

the C∗-algebra A to be separable.

Remark 3.1.4. In ourKK-theoretical computations on the cylinder, we work with unbounded

Kasparov modules. To this end we need the operator(M2VN + 1

)−1to be A-compact. Notice

that for an arbitrary Callias-type operator PV it is not true that(P 2V + 1

)−1is in KA(H0).

In the case when A = C, an easy spectral argument shows that this is the case when the

potential Φ goes to infinity at infinity. In the general case, we make a special choice of the

potential used to define MVN and in Section 4.2 we show that, with this choice,(M2VN +1

)−1is A-compact.

29

Remark 3.1.5. The advantage of formula (3.4) is that it reduces the computation of the

index of an elliptic operator on a noncompact manifold to the computation of the index

of an operator on the closed manifold N . Therefore, we can use the rich theory of elliptic

operators on closed manifolds to interpret the class indA PV . The application of this theorem

that we present in Chapter 5 is based on this fact.

3.2 Some properties of the index of twisted Callias-

type operators

In this section we establish some properties of the index class of twisted Callias-type oper-

ators. We use these properties to do the deformations and “cut-and-paste” constructions

needed in Section 3.3.

3.2.1 A sufficient condition for the vanishing of the A-index

Let M , E, V be as in Subsection 2.1 and let BV ∈ Diff1(M ;E ⊗ V ) be a twisted formally

self-adjoint Dirac-type operator.

Proposition 3.2.1. Suppose B2V is invertible with(B2V)−1 ∈ LA(H0, H2). Then the class

indABV vanishes.

Before proving this proposition, let us use it to deduce the following consequence.

Corollary 3.2.2. Let PV be the twisted Callias-type operator associated to an admissible

quadruple(Σ, D,Φ, V

)over a complete Riemannian manifold M . If Φ has empty essential

support (see Definition 2.3.1), then the class indA PV vanishes.

Proof. By Lemma 2.3.7, the operator P 2V + 1 is invertible and the inverse(P 2V + 1

)−1is in

LA(H0) ∩ LA(H0, H2). We write

P 2V ={

Id −(P 2V + 1

)−1}(P 2V + 1

). (3.5)

Since Φ has empty essential support, there exists a constant C > 0 such that〈(P 2V + 1

)s, s〉

0≥〈(

Φ2V − ‖[DV ,ΦV ]‖+ 1)s, s〉

0≥ (C + 1) 〈s, s〉0 , s ∈ H

2 .

(3.6)

30

Arguing as in the proof of Lemma 2.3.8 and Theorem 2.3.10, from Equations (3.5) and (3.6)

we deduce that P 2V is invertible with inverse (P2V )−1 in LA(H0, H2). Now the thesis follows

from Proposition 3.2.1.

In the remaining part of this subsection we prove Proposition 3.2.1. Under the hypoth-

esis of Proposition 3.2.1, by Theorem 2.3.6 it follows that the class indABV ∈ K0(A) is

represented by the bounded Kasparov module(H0, 1, BV

(B2V)−1/2)

(3.7)

for the pair of algebras (C, A). To show that the class defined by this triple vanishes, we

study the operator

F := BV(B2V)−1/2 ∈ LA(H0) .

Denote by F ∗ the adjoint of F .

Lemma 3.2.3. For all w ∈ H1, we have

F ∗w =(B2V)−1/2

BV w .

Proof. Fix u, v ∈ H0. We have〈(B2V)−1

u, v〉

0=〈(B2V)−1

u,B2V(B2V)−1

v〉

0=〈B2V(B2V)−1

u,(B2V)−1

v〉

0=〈u,(B2V)−1

v〉

0.

Therefore,(B2V)−1

is a self-adjoint element of LA(H0). By functional calculus,(B2V)−1/2

is

a self-adjoint element of LA(H0) as well. Fix w ∈ H1 and v ∈ H0. Using Theorem 2.2.7, we

obtain〈(B2V)−1/2

BVw, v〉

0=〈BVw,

(B2V)−1/2

v〉

0=〈w,BV

(B2V)−1/2

v〉

0= 〈w,Fv〉0 ,


3.2.2 Proof of Proposition 3.2.1

By [21, Proposition 3.28], it suffices to prove that the operator F is invertible. Fix an element

w ∈ H1. Then there exists u ∈ H0 such that w =(B2V)−1/2

u. From Lemma 3.2.3, we deduce

F ∗Fw =(B2V)−1/2

B2V(B2V)−1

u =(B2V)−1/2

u = w . (3.8)

31

Since H1 is dense in H0, Equation (3.8) implies that F ∗F = 1. Fix u ∈ H1. Since B2V is

invertible, u = BV x for some x ∈ H2. Therefore,

FF ∗u = BV(B2V)−1

B2V x = BV x = u . (3.9)

Since H1 is dense in H0, Equation (3.9) implies that FF ∗ = 1. It follows that F is invertible

with F−1 = F ∗, from which the thesis follows. �

3.2.3 A compact perturbation of the potential

Let M , E, V be as in Subsection 2.1 and let B0 ∈ Diff1(M ;E⊗V ) be a twisted formally self-

adjoint Dirac-type operator. Fix a self-adjoint potential Ψ ∈ C∞c (M ; End(E)) and consider

the operator

B1 := B0 + ΨV ,

where ΨV := Ψ ⊗ idV . Since Ψ is self-adjoint, B1 is also a twisted formally self-adjoint

Dirac-type operator.

Proposition 3.2.4. Suppose there exists a function f ∈ C∞c (M) such that B2j+f is invertible

with(B2j + f

)−1 ∈ LA(H0, H2), for j = 0, 1. Then B0 and B1 are invertible at infinity andtheir index classes coincide.

Before proving this proposition, we prove the following consequence.

Corollary 3.2.5. Let P1 and P2 be the twisted Callias-type operators associated respectively

to admissible quadruples(Σ, D,Φ1, V

)and

(Σ, D,Φ2, V

)over a complete Riemannian man-

ifold M . Suppose that

(i) the difference P2 − P1 is a compactly supported endomorphism of Σ⊗ V ;

(ii) there exists a compact set K that is an essential support for both, Φ1 and Φ2, with

respect to the pair(Σ, D

).

Then the classes indA P1 and indA P2 coincide.

32

Proof. By the hypothesis (i) and Proposition 3.2.4, it suffices to show that there exists a

function f ∈ C∞c (M) such that B2j + f is invertible with(B2j + f

)−1 ∈ LA(H0, H2), forj = 0, 1.

By the hypothesis (ii), there exist constants c1 and c2 satisfying

Φ2j(x)−∥∥[D,Φj](x)∥∥ ≥ cj , x ∈M \K , j = 1, 2 . (3.10)


f(x) −∥∥[D,Φ](x)∥∥ ≥ max (c1, c2) , x ∈ K . (3.11)

Arguing as in the proof of Theorem 2.3.10 (see Subsection ??), we conclude that f has the

wanted properties.

Our proof of Proposition 3.2.4 makes use of an integral representation due to Bunke. Let

BV be a twisted formally self-adjoint Dirac-type operator and let q : M → R be a smooth

function which is constant outside of a compact set. Suppose that B2V + q is invertible with(B2V + q

)−1in LA(H0, H2). By [12, Lemma 1.5], for all λ ≥ 0 the operator B2V + q + λ2 is

also invertible with(B2V + q + λ

2)−1 ∈ LA(H0, H2). Moreover, by [12, Lemma 1.8] we have

BV(B2V + q

)−1/2=

2

π

∫ ∞0

BV(B2V + q + λ

2)−1

dλ , (3.12)

where the integral converges in norm. The advantage of this formula is that the integrand

operator on the right hand side is easier to analyze than the operator on the left hand side.

Our analysis makes use of the following lemma.

Lemma 3.2.6 (Relllich lemma). Let ν ∈ C∞c (M). Then the multiplication operator ν :

H l → Hk is A-compact for every k < l.

This lemma provides a basic set of A-compact operators. In the C∗-algebra setting, it

was first proved in [18]. We use the formulation given in [12].

Let us introduce some notation. Let the operators B0, B1 and the function f be as in

the hypothesis of Proposition 3.2.4. For j = 0, 1 set

Fj := Bj(B2j + f

)−1/2and

Rj(λ) :=(B2j + f + λ

2)−1

.

33


We will show that F1 − F0 is A-compact. In fact, in this case the triples (H0, 1, F0) and

(H0, 1, F1) define the same class in KK(C, A) = K0(A) (see [4, Section 17.2]). By For-

mula (3.12), we have

F1 − F0 =2

π

∫ ∞0

{B1R1(λ)−B0R0(λ)} dλ ,

where the integral is norm-convergent. Therefore, it is enoug to show that the integrand

operator is A-compact, for all λ ≥ 0. We have

B1R1(λ)−B0R0(λ) = (B1 −B0)R1(λ) +B0 {R1(λ)−R0(λ)}

= ΨV R1(λ) +B0R0(λ)(B20 −B21

)R1(λ)

= ΨV R1(λ) +B0R0(λ)({B0,ΨV }+ Ψ2V

)R1(λ) .

Here, {B0,ΨV } = B0 ΨV + ΨV B0 is the anticommutator of B0 and ΨV . To complete

the proof, we will show separately that the operators ΨV R1(λ) and B0R0(λ)({B0,ΨV } +

Ψ2V

)R1(λ) are A-compact.

Let L ⊂M be a compact set whose interior contains the support of ΨV and let νL : M →

[0, 1] be a compactly supported function such that νL = 1 on L. Since ΨV is compactly

supported, it defines an operator ΨV ∈ LA(H2, H2). Since νL is compactly supported,

pointwise multiplication by νL defines an operator in KA(H2, H0) by Lemma 3.2.6. Moreover,

the operator ΨV R1(λ) is given by the composition

H0R1(λ)−−−−→ H2 ΨV−−−−−→ H2 νL−−−−−→ H0

so that it is in KA(H0).

It remains to analyze the operatorB0R0(λ)({B0,ΨV }+Ψ2V

)R1(λ). Since ΨV is compactly

supported, {B0,ΨV } + Ψ2V defines an operator in LA(H2, H1). Since νL is compactly sup-

ported, pointwise multiplication by νL defines an operator in KA(H1, H0) by Lemma 3.2.6.

Moreover, the operator B0R0(λ)({B0,ΨV }+ Ψ2V

)R1(λ) ∈ LA(H0) is given by the composi-

tion

H0R1(λ)−−−−→ H2

{B0,ΨV }+Ψ2V−−−−−−−−−−−→ H1 νL−−−−−→ H0 B0R0(λ)−−−−−−→ H0

so that it is A-compact. The proof is complete. �

34

3.2.5 A family of operators invertible at infinity

Let M , E, V be as in Subsection 2.1 and let BV ∈ Diff1(M ;E ⊗ V ) be a twisted for-

mally self-adjoint Dirac-type operator. Fix a uniformly bounded, self-adjoint potential

Ψ ∈ C∞(M ; End(E)) and consider the operator

Bt := BV + tΨV , t0 ≤ t ≤ t1 , (3.13)

where ΨV := Ψ ⊗ idV . Since Ψ is self-adjoint, each operator Bt is also a twisted formally

self-adjoint Dirac-type operator.

For j = 1, 2 and t0 ≤ t ≤ t1, let Hjt be the j-th Sobolev space defined by the operator Bt(see Subsection 2.3.3). We make the following assumption.

(F.1) For j = 1, 2 and for all s, t ∈ [t0, t1] the Sobolev spaces Hjs and Hjt are isomorphic as

Hilbert A-modules.

By (F.1), we denote by Hj the j-th Sobolev space defined by any of the operators Bt. We

also make the follwing assumptions.

(F.2) The operators {B2t }t0≤t≤t1 are uniformly invertible at infinity, i.e. there exists f ∈

C∞c (M) such that, for t0 ≤ t ≤ t1, the operator B2t + f is invertible with(B2t + f

)−1 ∈LA(H0, H2).

(F.3) the anticommutator {BV ,ΨV } = BV ΨV + ΨV BV defines an operator in LA(H1, H0).

Proposition 3.2.7. Suppose that Conditions (F.1)–(F.3) are satisfied. Then the class

indABt in K0(A) is independent of t.

Before proving this proposition, we deduce three consequences that will be used in Sec-

tion 3.3.

Corollary 3.2.8. Let PV be the twisted Callias-type operator associated to an admissible


)over a complete Riemannian manifold M . Assume that the endo-

morphism Φ is constant outside a compact set. Fix λ ≥ 1. Then the endomorphism λΦ is

admissible for the pair(Σ, D

). Moreover, if P λV is the twisted Callias-type operator associated

to the admissible quadruple(Σ, D, λΦ, V

), then indA P

λV = indA PV .

35

Proof. Let K be an essential support of Φ. Then there exists c > 0 such that

Φ2(x)− ‖[D,Φ](x)‖ ≥ c , x ∈M \K .

For 1 ≤ t ≤ λ, we have

(tΦ)2(x)− ‖[D, tΦ](x)‖ = t(tΦ2(x)− ‖[D,Φ](x)‖

)≥ t c , (3.14)

for all x ∈M \K. For t = λ, this inequality implies that λΦ is admissible for(Σ, D

)so that

the operator P λV is well-defined.

It remains to show that the classes indA PλV and indA PV coincide. Consider the family

of endomorphisms Φt := tΦ, for 1 ≤ t ≤ λ. Inequality (3.14) implies that each Φt is

admissible for the pair(Σ, D

). Denote by Qt the twisted Callias-type operator associated

to the admissible quadruple (Σ, D, tΦ, V ). Note that Qt is of the form (3.13):

Qt = DV L + t(ΦK

), 1 ≤ t ≤ λ ,

where L and K are the matrices defined by

L :=

0 11 0

J := 0 −i

i 0

. (3.15)Since Φ is constant outside of a compact set, Condition (F.1) and (F.3) are satisfied. From

Theorem 3.2.7, to prove the thesis it suffices to show that Condition (F.2) holds.


f(x) = λ max{‖[D,Φ](z)‖ : z ∈ K

}+ c , x ∈ K.

Then

t2Φ2(x) + f(x)− t ‖[D,Φ](x)‖ ≥ f(x)− λ ‖[D,Φ](x)‖ ≥ c , x ∈ K . (3.16)

From (3.14), we deduce

t2Φ2(x)+f(x)− t ‖[D,Φ](x)‖ ≥ t2Φ2(x)− t ‖[D,Φ](x)‖ ≥ c , x ∈M \K . (3.17)

Arguing as in the proof of Theorem 2.3.10 (Subsection ??), from (3.16), (3.17) we get〈(Q2t + f

)s, s〉

0≥〈(t2Φ2 + f − t ‖[D,Φ]‖

)s, s〉

0≥ 〈s, s〉0 , s ∈ H

2 .

Now Condition (F.2) follows from Theorem 2.3.9.

36

Corollary 3.2.9. Let P1 and P2 be the twisted Callias-type operators associated respectively

to admissible quadruples(Σ, D,Φ1, V

)and

(Σ, D,Φ2, V

)over a complete Riemannian man-

ifold M . Suppose that

(i) the endomorphisms Φ1 ,Φ2 are constant outside a compact set;

(ii) for 0 ≤ t ≤ 1, the endomorphism Φt := tΦ2 + (1 − t)Φ1 is admissible with essential

support independent of t.

Then indA P1 = indA P2.

Proof. By hypothesis, {Φt}0≤t≤1 is a family of uniformly bounded admissible endomorphisms

for the pair(Σ, D

). Denote by Ut the twisted Callias-type operator associated to the admis-

sible quadruple (Σ, D,Φt, V ). Note that Ut is of the form (3.13):

Ut =(DV L + Φ1K

)+ t (Φ2 − Φ1) K , 0 ≤ t ≤ 1 ,

where L and K are the matrices defined in (3.15). By the hypothesis (i), the endomorphism

Φ2 − Φ1 is constant outside a compact set. Hence, the family of operators {Ut}0≤t≤1 satisfy

Conditions (F.1) and (F.3). By Proposition 3.2.7, to prove the thesis it suffices to show

that Condition (F.2) holds.

By the hypothesis (ii), there exist a compact set K ⊂M and a constant c > 0 such that

Φ2t (x) − ‖[D,Φt](x)‖ ≥ c , x ∈M \K . (3.18)


f(x) = c + maxz∈K

(‖[D,Φ1](z)‖+ ‖[D,Φ2](z)‖

), x ∈ K . (3.19)

From (3.22), we have

Φ2t (x) + f(x) − ‖[D,Φt](x)‖ ≥ Φ2t (x)− ‖[D,Φt](x)‖ ≥ c , x ∈M \K . (3.20)

By (3.23), we deduce

Φ2t (x) + f(x)− ‖[D,Φt](x)‖ ≥ f(x)− ‖[D,Φt](x)‖ ≥ c , x ∈ K. (3.21)

37


〈(U2t + f

)s, s〉

0≥〈(

Φ2t + f − ‖[D,Φt]‖)s, s〉

0≥ c 〈s, s〉0 , s ∈ H

2 .


Corollary 3.2.10. Let(Σ, D,Φ, V

)be an admissible quadruple over a complete Riemannian

manifold M . Suppose that Π ∈ C∞(M ; End(Σ)) is a self-adjoint potential such that

(i) Π is constant outside a compact set;

(ii) for 0 ≤ t ≤ 1, the endomorphism Φ is admissible for the pair(Σ, D + tΠ

), with

essential support independent of t.

If P0, P1 are the Callias-type operators associated respectively to the quadruples(Σ, D,Φ, V

)and

(Σ, D + Π,Φ, V

), then the classes indA P1 and indA P2 coincide.

Proof. LetWt be the twisted Callias-type operator associated to the quadruple (Σ, D + tΠ,Φ, V ).

Note that Wt is of the form (3.13):

Wt =(DV L + ΦK

)+ t

(ΠL), 0 ≤ t ≤ 1 ,

where I and J are the matrices defined in (2.38). By the hypothesis (i), the endomorphism

Π is constant outside a cmpact set. Hence, the family of operators {Wt}0≤t≤1 satisfy Con-

ditions (F.1) and (F.3). By Proposition 3.2.7, to prove the thesis it suffices to show that

Condition (F.2) holds.

By the hypothesis (ii), there exist a compact set K ⊂M and a constant c > 0 such that

Φ2(x) − ‖[D + tΠ,Φ](x)‖ ≥ c , x ∈M \K , 0 ≤ t ≤ 1 . (3.22)


f(x) = c + maxy∈K‖[D,Φ1](y)‖ + max

z∈K‖[D,Φ2](z)‖ , x ∈ K . (3.23)

From (3.22), we have

Φ2(x) + f(x) − ‖[D + tΠ,Φt](x)‖ ≥ Φ2(x) − ‖[D + tΠ,Φ](x)‖ ≥ c , x ∈M \K .

(3.24)

38

By (3.23), we deduce

Φ2(x) + f(x) − ‖[D+ tΠ,Φt](x)‖ ≥ f(x)−‖[D+ tΠ,Φ](x)‖ ≥ c , x ∈ K. (3.25)


〈(W 2t + f

)s, s〉

0≥〈(

Φ2 + f − ‖[D + tΠ,Φ]‖)s, s〉

0≥ c 〈s, s〉0 , s ∈ H

2 .


The remaining part of this subsection is devoted to proving Proposition 3.2.7. By Con-

dition (F.2) and Theorem 2.3.6, the class indABt ∈ K0(A) is represented by the bounded

Kasparov module (H0, 1, Ft), where

Ft := Bt(B2t + f

)−1/2(3.26)

and where f is any compactly supported smooth function such that(B2t +f

)−1 ∈ LA(H0, H2)for t0 ≤ t ≤ t1. In order to study the operator Ft, we use the integral representation (3.12).

For this reason, we also consider the operator

Rt(λ) :=(B2t + f + λ

2)−1

.

Notice that, by [12, Lemma 1.5], Rt(λ) ∈ LA(H0, H2) ∩ LA(H0).

Lemma 3.2.11. For all s, t ∈ [t0, t1], the difference Bs −Bt is a uniformly bounded bundle

map and we have

‖Bs − Bt‖∞ ≤ |s− t| ‖Ψ‖∞ . (3.27)

Proof. From (3.13), we deduce

Bs − Bt = (s− t)(Ψ⊗ idV

),


Lemma 3.2.12. There exists a constant d > 0 such that

‖Rt(λ)‖B(H0) ≤ (d+ λ2)−1 , t0 ≤ t ≤ t1 . (3.28)

39

Proof. By [12, Lemma 1.5], we have

‖Rt(λ)‖B(H0) ≤(dt + λ

2)−1

, (3.29)

where

0 < dt := inf{(‖Btu‖20 +

∥∥√fu∥∥20

): u ∈ H2, ‖u‖0 = 1

}.

Notice that the positivity of dt is not a trivial fact and is guaranteed by [12, Lemma 1.4].

From (3.29) we deduce that Inequality (3.28) holds with

d := inf {dt : t0 ≤ t ≤ t1} .

Finally, from Lemma 3.2.11, it follows that {dt}t0≤t≤t1 varies continuously so that d is strictly

positive, which concludes the proof.

Lemma 3.2.13. There exists a constant c1 > 0 such that

‖BtRt(λ)‖B(H0) ≤ c1(d+ λ2

)−1/2, t0 ≤ t ≤ t1 . (3.30)

Proof. Notice that

B2t Rt(λ) =[Rt(λ)

−1 − f − λ2]Rt(λ) = Id −

(f + λ2

)Rt(λ) .

Using (3.28) and the previous equality, we obtain

∥∥B2t Rt(λ)∥∥B(H0) ≤ 1 + ∥∥(f + λ2)Rt(λ)∥∥B(H0) ≤ 1 + ‖f‖∞ + λ2d+ λ2 ≤ c 2 , (3.31)where c is a positive constant such that c 2 ≥ max

(2, 1 + ‖f‖∞ d−1

). By (3.28) and (3.31),

for u ∈ H0 we get

‖BtRt(λ)u‖20 =∣∣ 〈BtRt(λ)u,BtRt(λ)u〉0 ∣∣A = ∣∣ 〈Rt(λ)u,B2t Rt(λ)u〉0 ∣∣A

≤ ‖Rt(λ)u‖0∥∥B2t Rt(λ)u∥∥0 ≤ c 2 (d+ λ2)−1 ‖u‖20 ,


Lemma 3.2.14. There exists a constant c2 > 0 such that

‖{B,Ψ}Rt(λ)‖LA(H0) ≤ c2(d+ λ2

)−1/2, λ ≥ 1, t0 ≤ t ≤ t1 . (3.32)

40

Proof. Fix t ∈ [t0, t1], and u ∈ H0. By Lemma 3.2.12 and Lemma 3.2.13, we have

‖Rt(λ)u‖1 ≤ ‖Rt(λ)u‖0 + ‖BtRt(λ)u‖0

≤(d+ λ2

)−1 ‖u‖0 + c1 (d+ λ2)−1/2 ‖u‖0 ≤ (d−1/2 + c1) (d+ λ2)−1/2 ‖u‖0 .Hence, using Condition (F.3) we obtain

‖{B,Ψ}Rt(λ)u‖0 ≤ ‖{B,Ψ}‖LA(H1,H0)(d−1/2 + c1

) (d+ λ2

)−1/2 ‖u‖0 ,from which the thesis follows.


We will show that the family{Ft}t0≤t≤t1

defined by (3.26) is continuous in LA(H0). By (3.12),

we deduce

Fs − Ft =2

π

∫ ∞0

{BsRs(λ)−BtRt(λ)} dλ . (3.33)

We now analyze the integrand operator on the right hand side of (3.33). Note that

B2t = B2V + t {BV ,ΨV } + t2Ψ2V

so that

B2t − B2s = (t− s) {BV ,ΨV } +(t2 − s2

)Ψ2V .

It follows that

BsRs(λ)−BtRt(λ) =(Bs −Bt

)Rs(λ) +Bt

(Rs(λ)−Rt(λ)

)=(Bs −Bt

)Rs(λ) +BtRt(λ)

(B2t −B2s

)Rs(λ)

= (s− t){

ΨV Rs(λ) − BtRt(λ) {BV ,ΨV }Rs(λ) − BtRt(λ) (s+ t) Ψ2V Rs(λ)}.

(3.34)

Using Lemma 3.2.12, Lemma 3.2.13, and Lemma 3.2.14, from (3.33) and (3.34) we deduce

that there exists a constant c3 > 0 such that

∥∥Fs − Ft∥∥B(H0) ≤ |s− t| c3 2π∫ ∞

0

dλ

d+ λ2,

from which the thesis follows. �

41

3.2.7 The opposite potential

In this subsection we study how the index of a Callias-type operator changes if we substitute

a potential with its opposite.

Proposition 3.2.15. Let PV be the twisted Callias-type operator associated to an admissible


)over a complete Riemannian manifold M . Then the endomorphism

−Φ is also admissible for the pair(Σ, D

). Moreover, if −PV is the twisted Callias-type

operator associated to the admissible quadruple(Σ, D,−Φ, V

), then indA(−PV ) = − indA PV .

As an application of this proposition, we deduce the following consequence.

Corollary 3.2.16. Let PV be a twisted Callias-type operator over a compact manifold M .

Then the class indA PV vanishes.

Proof. Let(Σ, D,Φ, V

)be the admissible quadruple over M defining the operator PV . Con-

sider the family of admissible endomorphisms

Φt := t (−Φ) + (1− t) Φ = Φ (1− 2t) , 0 ≤ t ≤ 1 .

For t = 0, we obtain the initial quadruple(Σ, D,Φ, V

)and, for t = −1, we obtain the

admissible quadruple(Σ, D,−Φ, V

), with associated twisted Callias-type operator −PV .

From Corollary 3.2.9 and Proposition 3.2.15, we deduce

indA PV = indA(−PV ) = − indA PV ,


3.2.8 The Fredholm picture

In order to prove Proposition 3.2.15, we make use of the “Fredholm picture” of the A-index

of twisted Callias-type operators.

Let the operator PV , the quadruple(Σ, D,Φ, V

)and the manifold M be as in the hypoth-

esis of Proposition 3.2.15. From Theorem 2.3.10, the class indA PV in K0(A) is represented

by the bounded Kasparov moduleH0(Σ⊗ V )⊕H0(Σ⊗ V ), 1, 0 F−

F+ 0

, (3.35)

42

where 1 is complex scalar multiplication, F± =(DV ± iΦV

)(P 2V + f

)−1/2and f ∈ C∞c (M)

is as in Definition 2.3.5.

Let HA be the standard countably generated Hilbert module over A (see [35, Exam-

ple 2.2]) and consider the operator F ⊕ idHA : HA → HA. Here, we use the Kasparov sta-

bilization theorem ([41, Theorem 15.4.6]) to identify H0 ⊕HA and HA. By Atkinson-Mingo

Theorem ([41, Theorem 17.1.6]) and [12, Lemma 1.11], there exist A-compact operators R±

such that Ker(F± ⊕ idHA +R±) are finitely generated projective Hilbert A-modules. Define

the class

Mindex(F+) :=[

Ker(F+ ⊕ idHA +R+

)]−[

Ker(F− ⊗ idHA +R−

)]∈ K0(A) , (3.36)

where[

Ker(F± ⊕ idHA +R±

)]are the isomorphism classes of Ker

(F± ⊕ idHA +R±

). By

[41, Corollary 17.2.5], the class Mindex(F+) is independent of the choice of the A-compact

operators R± such that Ker(F± ⊕ idHA +R±

)are finitely generated projective Hilbert A-

modules. By [4, Section 17.5], the classes Mindex(F ) and indA PV coincide.


The class indA(−PV ) is represented by the bounded Kasparov moduleH0(Σ⊗ V )⊕H0(Σ⊗ V ), 1, 0 F+

F− 0

,

where we use the same notation as in (3.35). Using the Fredholm picture, we finally obtain

indA(−PV ) = Mindex(F−) =[

Ker(F− ⊗ idHA +R−

)]−[

Ker(F+ ⊕ idHA +R+

)]= −Mindex(F+) = − indA PV ,

where R± denote the same objects as in (3.36). �

3.3 Reduction to the cylinder

This section is devoted to the proof of the first step of Theorem 3.1.1 (see Remark 3.1.3).

We start with a twisted Callias-type operator PV associated to an admissible quadruple

43

(Σ, D,Φ, V ) over a complete oriented Ri

c -algebras a dissertation submitted tocj... · 2019. 2. 13. · a dissertation submitted to the...

Documents