A Metric in the Space of Spectral Triples

Home , Gelfand representation

Utrecht University

Master Thesis

A metric in the space of spectral triples

Supervisor: Author: Gunther Cornelissen Florian Kluck Second Reader: Marius Crainic

January 28, 2014 ii Contents

1 Context 1 1.1 The Gelfand representation theorem ...... 1 1.2 The Gromov-Hausdorﬀ distance ...... 3

2 Basic Theory 9 2.1 C*-modules ...... 9 2.2 Operator spaces ...... 12

3 Connections 19 3.1 Connections ...... 19

4 Spectral Triples 25 4.1 Spectral Triples ...... 25 4.2 C1-modules ...... 26 4.3 Unbounded selfadjoint operators on C1-modules ...... 27 4.4 The gap metric ...... 29 4.5 Morphisms of spectral triples ...... 30

5 A correspondence between circles 33

Bibliography 39

Acknowledgements

This thesis was written as part of the master’s program Mathematical Sciences at Utrecht University, under the supervision of Gunther Cornelissen. I would like to express my thanks to him for helping me ﬁnd a suitable subject, for giving me a good introduction in the subject and for helping me when I got stuck in the theory. I also would like to thank Bram Mesland for answering my questions when I didn’t understand some part of his theory. He did this with both great speed and clarity. I also would like to thank Gunther Cornelissen and Bram Mesland together for allowing me to read along while they developed a new theory, thus allowing me to see how mathematics is created, whereas up until this point, I have only seen mathematics that was already fully developed. I have now seen that mathematics can really be an exploration into the unknown, where sometimes you make guesses which turn out to be incorrect and sometimes you somehow stumble upon some unexpected, but nice result.

iii iv CONTENTS

Introduction

The Gelfand representation theorem yields a isometric *-isomorphism between any commutative C*-algebra and the space of continuous functions on some Hausdorff space. This allowed people to study Hausdorff spaces by studying commutative C*-algebras. Later, people also started to study noncommutative C*-algebras as if they belonged to “noncommutative Hausdorff spaces”. Thus, noncommutative geometry was born. In 1996, Alain Connes introduced the spectral triple in [5]. A spectral triple encodes the information of a spin manifold in a way that allows for a noncommutative generalization. In chapter 3 of [16] a set of axioms are listed that ensure that any spectral triple satisfying them arises from a spin manifold. Apart from that, in 1981, Misha Gromov introduced a metric on the space of compact metric spaces modulo isometry, called the Gromov-Hausdorff distance. Some properties of compact metric spaces are preserved by taking a Gromov- Hausdorff limit, and a convenient property is that all compact Hausdorff spaces can be obtained as the limit of a finite space (see example 7.4.9 in [9]). With these ideas in mind, Gunther Cornelissen and Bram Mesland are working on a metric space of spectral triples. This starts by defining a correspondence between spectral triples (see [13]) and continues by defining the length of a correspondence ([8]). The distance between two spectral triples is then defined as the infimum of the lengths of all correspondences between them.

Survey and Notation

In general A denotes a C∗-algebra, where an operator algebra would be denoted by A. Similarly, E denotes a C∗-module, and E denotes an operator module. 1 In [13], the notation E and A1 is used for specifically defined submodules resp. subalgebras of C∗-modules E , resp. C∗-algebras A . Since the former objects are the only examples of operator modules and operator algebras used in this thesis, we will denote them by just E and A. In this thesis, I will follow the work of Gunther Cornelissen and Bram Mes- land in [8], where they define the concept of a correspondence between spectral triples, and the length of such a correspondence. This then gives rise to a definition of a distance between two spectral triples as the infimum of the length of all possible correspondences between the two spectral triples. In the first chapter, the context will be given for this theory. One could also regard this as some sort of motivation for the subject. In section 1.1, the Gelfand representation theorem will be discussed. This theorem states that there is a isometric *-isomorphism between any commutative C*-algebra and the space of continuous functions on some Hausdorff space. It follows that two commutative C*-algebras are isometrically *-isomorphic if and only if the corresponding Hausdorff spaces are homeomorphic. This enables one to study the topology of manifolds in an algebraic manner. Also, the first step towards noncommutative geometry is to study a noncommutative C*-algebra as if it were a topological space. In section 1.2, the Gromov-Hausdorff distance will be introduced, which is a distance function between compact Hausdorff spaces. This serves to illustrate the idea of measuring the distance between spaces. CONTENTS v

The second chapter discusses the theoretic background, introducing all the objects necessary to define spectral triples and correspondences. In section 2.1, the theory of C*-modules will be reviewed succinctly. C*-modules are a generalization of Hilbert spaces, and are used to form correspondences between spectral triples, both of which will be discussed in section 4.1. In section 2.2, an introduction to the theory of operator spaces will be given. An operator space X can be viewed as a Banach space, where each point is an operator on a Hilbert space. As such, an operator space X comes with a norm on the space of n × n-matrices with entries in X. In chapter 3.1, the concept of a connection is introduced. Given operator spaces X, Y and an operator D on Y , that is not necessarily left linear with respect to the action of an algebra on Y , a connection is used to extend the operator D to an operator on the tensor product X ⊗A Y . In sections 4.1 and 4.5 the key concepts of this thesis will be introduced, namely spectral triples and correspondences between them. A spectral triple is a generalization of spin manifolds to noncommutative geometry, and consist of a triple of an algebra, faithfully represented on a Hilbert space and an operator on that Hilbert space. A correspondence is a way to transform one spectral triple into another. In section 4.2 a class of operator modules called C1-module will be defined an in 4.3 unbounded operators on C1-modules will be discussed. A class of suffi- ciently well behaved operators are the regular operators, defined in this section. Finally, the gap distance between two regular operators will be introduced in 4.4. Then, a definition for the length of a correspondence will be given and this will be used to construct a distance between spectral triples as the infimum of the lengths all possible correspondences between them. In chapter 5, we will see an example of a correspondence from one circle to another, the second of which has a radius that is an integer multiple of the other. This will be discussed in detail and serves to illustrate the concept of a correspondence. vi CONTENTS Chapter 1

Context

In this chapter, we will review the Gelfand representation theorem and the Gromov-Hausdorff distance. The Gelfand representation theorem provides a useful link between C*-algebras and compact Hausdorff spaces, allowing for an algebraic study of topological properties. This serves to illustrate the link between algebra and topology, of which spectral triples are a more advanced example. The Gromov-Hausdorff distance serves to illustrate what a distance between certain kind of spaces looks like. This should make the length function of correspondences between spectral triples more intuitive.

1.1 The Gelfand representation theorem

Most of the proofs in this section are based on paragraph VII.8 in [6]. Through- out the next section, all C*-algebras are assumed to be unital.

Lemma 1.1.1 ([6] Prop.VIII.1.11.e). If A is a C*-algebra, and a ∈ A is such that a = a∗, then ||a|| = r(a), where r(a) denotes the spectral radius of a.

Proof. By the axioms of a C*-algebra, ||a||2 = ||a∗a|| = ||a2||. By induction n 1 1 2 n n it holds for each n ≥ 1, that ||a || 2 = ||a||. Then r(a) = limn ||a || n = 2k 1 limk ||a || 2k = ||a||.

Lemma 1.1.2 ([6] Thm.VII.8.1). If A is a Banach algebra that is also a division ring, then A =∼ C. Proof. For a ∈ A , denote by σ(a) the spectrum of a, which is nonempty. For λ ∈ σ(a), a − λ is not invertible. Since A is a division ring, a − λ = 0, and hence a = λ.

Lemma 1.1.3 ([6] Prop.VII.8.2). If M is a maximal ideal in a commutative C*-algebra A , then there exists a nonzero homomorphism h : A −→ C such that M = ker h.

Proof. If M is a maximal ideal, then it is closed, so A /M is again a C*-algebra with unit. Let π : A −→ A /M denote the quotient map and let a ∈ A be an element such that π(a) is not invertible. Then π(A a) is a proper ideal in A /M and I := π−1(π(A a)) is a proper ideal in A , containing M. Since M is

1 2 CHAPTER 1. CONTEXT maximal, I = M. Now, a ∈ I, so π(a) ∈ π(M) = {0}. So A /M is a ﬁeld. By lemma 1.1.2, A /M =∼ C. Let j : A/M −→ C be an isomorphism, and deﬁne h = j ◦ π. Then h is a homomorphism such that M = ker h.

Given a C*-algebra A , deﬁne Σ, the maximal ideal space of A ,

Σ := {h : A −→ C | h is a nonzero homomorphism} (1.1) endowed with the weak*-topology that it has as a subset of the dual space of A , i.e. the topology generated by the semi-norms {pa|a ∈ A }, where pa(h) := |h(a)|.

Lemma 1.1.4 ([6] Prop.VII.8.4). If A is a commutative C*-algebra and h : A −→ C is a nonzero homomorphism, then khk = 1.

Proof. Let a ∈ A and suppose towards a contradiction that |h(a)| > kak. a a Because k h(a) k < 1, the element 1 − h(a) is invertible. Denote its inverse by a a a h(a) b. Then 1 = b(1 − h(a) ) = b − b h(a) and h(b − b h(a) ) = h(b) − h(b) h(a) = 0. A contradiction because h(1) = 1. Hence |h(a)| ≤ kak and khk ≤ 1. Because h(1) = 1, khk = 1.

Lemma 1.1.5 ([6] Thm.VII.8.6). If A is commutative, then Σ is a compact Hausdorﬀ space.

Proof. By the previous proposition, Σ is contained in the unit ball of the dual space of A . Since Alaoglu’s Theorem (see [6] Thm.V.3.1), states that the unit ball of the dual space of A is compact in the wk*-topology, it suf- ﬁces to show that Σ is closed in the wk*-topology. Suppose {hi} is a net in Σ and h is an element of the unit ball such that hi → h. Then h(ab) = limi hi(ab) = limi hi(a)hi(b) = h(a)h(b), so h is a homomorphism. Because h(1) = limi hi(1) = 1, it is nonzero and hence h ∈ Σ.

Lemma 1.1.6 ([6] Thm.VII.8.6). For a ∈ A , deﬁne Σ(a) := {h(a) | h ∈ Σ}. If A is commutative, then Σ(a) = σ(a). Proof. For λ ∈ Σ(a), there exists h ∈ Σ such that h(a) = λ. But then h(a−λ) = h(a) − λ = 0. Hence a − λ ∈ ker h. So λ ∈ σ(a). Conversely, for λ ∈ σ(a), a − λ is not invertible, which implies that (a − λ)A is a proper ideal. Let M be a maximal ideal containing (a − λ)A , and let h ∈ Σ be such that M = ker h. Then 0 = h(a − λ) = h(a) − λ. Hence h(a) = λ and λ ∈ Σ(a). So Σ(a) = σ(a).

Theorem 1.1.7 ([6] Thm.VIII.2.1). If A is commutative, then the map γ : A −→ C(Σ) : a 7→ aˆ, where aˆ(h) := h(a), is an isometric *-isomorphism.

Proof. To see that γ maps A into the continuous functions on Σ, suppose that ∗ hi → h in (Σ, wk ). Thena ˆ(hi) = hi(a) → h(a) =a ˆ(h). To see that γ is a *-homomorphism, note that abb (h) = h(ab) = h(a)h(b) = ∗ ∗ ∗ (bab)(h) and that ab (h) = h(a ) = h(a) = (ba) (h). For a selfadjoint element a, the following holds:

||aˆ|| = sup |h(a)| = sup |λ| = sup |λ| = r(a) = ||a|| (1.2) h∈Σ λ∈Σ(a) λ∈σ(a) 1.2. THE GROMOV-HAUSDORFF DISTANCE 3

For a not necessarily selfadjoint element x, the previous can be used as follows:

2 ∗ ∗ 2 ||x|| = ||x x|| = ||xdx|| = || |xb| || = ||xb|| (1.3) Hence, γ is isometric. Since γ is isometric, its range is closed. By an easy application of the Stone- Weierstrass theorem, it follows that γ is surjective.

The previous section can be summarized as follows:

Theorem 1.1.8 ([6] Cor.VIII.2.2). If A is a commutative C*-algebra with unit, then there exists a compact Hausdorﬀ space X, unique up to homeomorphism, such that A =∼ C(X).

Proof. The only statement we have not proven is that the space X is unique up to homeomorphism. This is theorem VII.8.7 in [6].

This theorem allows us to study compact Hausdorﬀ spaces by studying C*- algebras.

1.2 The Gromov-Hausdorﬀ distance

The Gromov-Hausdorff distance is defined on the space of all compact metric spaces. At the end of this section, we will show that the space of all compact metric spaces up to isometry, and with the Gromov-Hausdorff distance, is itself a metric space. A more complete treatment of this can be found in [9]. We begin by defining a distance between two subsets of a single metric space, called the Hausdorff distance.

Definition 1.2.1. The Hausdorff distance between two subsets X, Y of a metric space Z is defined to be: dH (X,Y ) = max sup inf d(x, y), sup inf d(x, y) . x∈X y∈Y y∈Y x∈X When we want to emphasize in which surrounding space (Z, d) we calculate the Hausdorff distance, we will denote it by d(Z,d)(X,Y ).

Definition 1.2.2. The Gromov-Hausdorff distance between two metric spaces X, Y is defined to be:

dGH (X,Y ) = inf d(Z,d)(f(X), g(Y )), (Z,d),f,g where the inﬁmum is taken over all metric spaces (Z, d) and isometric embeddings f : X → Z, g : Y → Z.

There are many metric spaces in which two spaces can be embedded. For- tunately, as the next proposition shows, it is possible to restrict the attention to the disjoint union of X and Y , denoted by X t Y , with all possible metrics extending the ones of X and Y . 4 CHAPTER 1. CONTEXT

Proposition 1.2.3 ([9] Remark 7.3.12). If the inﬁmum on the left hand side is taken over all metric spaces Z and isometric embeddings f : X → Z, g : Y → Z and the inﬁmum on the right hand side is taken over all metrics on X t Y extending the ones on X and Y , then the following holds:

inf d(Z,d)(f(X), g(Y )) = inf d(XtY,d)(X,Y ). (1.4) (Z,d),f,g d

Proof. Since (X t Y, d) is a metric space in which X and Y can be isometrically embedded, the right-hand side of equation 1.4 is at least as large as the left-hand side.

Conversely, suppose we are given a metric space (Z, dZ ) and isometric embeddings f : X → Z and g : Y → Z. If we would extend the metrics on X and Y to X t Y by deﬁning d(x, y) = dZ (f(x), g(y)), we would get only a premetric if f(X) ∩ g(Y ) 6= ∅. So we deﬁne d(x, y) = dZ (f(x), g(y)) + δ, where δ is a positive constant. With this metric it holds that d(XtY,d)(X,Y ) ≤ d(Z,dZ )(f(X), g(Y )) + δ. Since this can be done for all positive δ, it holds that the left-hand side of 1.4 is at least as large as the right-hand side.

Definition 1.2.4. A function d : X × X −→ R+ ∪ {∞} is called a premetric if it satisfies all requirements of being a metric, except d(x, y) = 0 ⇒ x = y. So, it is a metric that allows different points to have zero distance between them. In [9], the terminology semimetric is used for a premetric.

Proposition 1.2.5 ([9] Prop.7.3.16). The Gromov-Hausdorﬀ distance deﬁnes a premetric on the space of all compact metric spaces.

Proof. Positive definiteness follows from the same property of the Hausdorff distance. Symmetry follows from the definition. It only remains to show that the Gromov-Hausdorff distance satisfies the triangle inequality. Let X, Y , and Z be compact metric spaces and let dXY be a metric on X t Y that extends the metrics on X and Y , and let dYZ be the same for the spaces Y and Z. Extend the metrics on X and Z to X t Z by defining for x ∈ X and z ∈ Z:

dXZ (z, x) = dXZ (x, z) := inf {dXY (x, y) + dYZ (y, z) + δ} . y∈Y

Where δ is again a positive constant, to ensure that we get a metric rather than just a premetric. Being the sum of two positive definite functions, dXZ is also positive definite and by definition it is symmetric. The triangle inequality follows from the following inequality, with x, x0 ∈ X and z ∈ Z:

dXZ (x, z) = inf {dXY (x, y) + dYZ (y, z) + δ} y∈Y 0 0 ≤ inf {dXY (x, x ) + dXY (x , y) + dYZ (y, z) + δ} y∈Y 0 0 = dXZ (x, x ) + dXZ (x , z), and a similar inequality where the x0 is replaced by an element z0 ∈ Z. Also, we need the following inequality for x, x0 ∈ X and z ∈ Z, obtained by application of 1.2. THE GROMOV-HAUSDORFF DISTANCE 5

the triangle inequality, ﬁrst for dYZ , followed by a double application for dXY :

0 0 0 0 d(x, z) + d(z, x ) = inf {dXY (x, y) + dYZ (y, z) + dXY (x , y ) + dYZ (y , z) + 2δ} y,y0∈Y 0 0 0 ≥ inf {dXY (x, y) + dYZ (y, y ) + dXY (x , y ) + 2δ} y,y0∈Y 0 0 0 = inf {dXY (x, y) + dXY (y, y ) + dXY (x , y ) + 2δ} y,y0∈Y 0 ≥ inf {dXY (x, x ) + 2δ} y,y0∈Y ≥ d(x, x0).

The equality in the middle holds because dXY and dYZ both are extensions of dY . Similarly:

0 0 0 0 d(z, x) + d(x, z ) = inf {dYZ (z, y) + dXY (y, x) + dXY (x, y ) + dYZ (y , z ) + 2δ} y,y0∈Y 0 0 0 ≥ inf {dYZ (z, y) + dXY (y, y ) + dYZ (y , z ) + 2δ} y,y0∈Y 0 0 0 = inf {dYZ (z, y) + dYZ (y, y ) + dYZ (y , z ) + 2δ} y,y0∈Y 0 ≥ inf {dYZ (z, z ) + 2δ} y,y0∈Y 0 ≥ dZ (z, z ).

Hence dXZ is a metric. It follows from the definition of dXZ that, when taking the Hausdorff distance with respect to these metrics, the following identity holds: dH (X,Z) ≤ dH (X,Y ) + dH (Y,Z). Taking the infimum over all metrics dXY , dYZ yields the triangle inequality for the Gromov-Hausdorff distance

A subset S ⊂ X of a compact metric space is called an ε-net, if for every point x ∈ X there is a point y ∈ S such that d(x, y) < ε. Given a function f : X → Y between two compact metric spaces, we deﬁne the distortion of f, denoted by dist(f), by

dist(f) := sup |d(x, x0) − d(f(x), f(x0))| x,x0

Deﬁnition 1.2.6. A function f : X → Y is called an ε- isometry if f(X) is an ε-net in Y and dist(f) < ε.

An ε-isometry f : X −→ Y can be used to deﬁne a metric d on the disjoint union of X and Y , by extending the metrics on X and Y to: n ε o d(y, x) = d(x, y) := inf d(x, z) + d(f(z), y) + ∀x ∈ X ∀y ∈ Y. z∈X 2

Lemma 1.2.7. The function d deﬁned above is a metric

Proof. Being the sum of two positive definite functions, d is also positive definite. It is symmetric by definition. The proof for the triangle inequality splits into 6 CHAPTER 1. CONTEXT several cases, first for x0 ∈ X: n ε o d(x, y) = inf d(x, z) + d(f(z), y) + z∈X 2 n ε o ≤ inf d(x, x0) + d(x0, z) + d(f(z), y) + z∈X 2 n ε o = d(x, x0) + inf d(x0, z) + d(f(z), y) + z∈X 2 = d(x, x0) + d(x0, y), where the inequality on the second line follows from the triangle inequality for the metric on X. Similarly, for y0 ∈ Y : n ε o d(x, y) = inf d(x, z) + d(f(z), y) + z∈X 2 n ε o ≤ inf d(x0, z) + d(f(z), y0) + d(y0, y) + z∈X 2 n ε o = inf d(x0, z) + d(f(z), y0) + + d(y0, y) z∈X 2 = d(x, y0) + d(y0, y).

The more interesting cases is where we use that dist(f) < ε:

d(x, y) + d(y, x0) = inf d(x, z) + d(f(z), y) + d(x0, z0) + d(f(z0), y) + ε z,z0∈X ≥ inf d(x, z) + d(f(z), f(z0)) + d(x0, z0) + ε z,z0∈X ≥ inf d(x, z) + d(z, z0) + d(x0, z0) z,z0∈X ≥ d(x, x0), and similarly:

d(y, x) + d(x, y0) = inf {d(x, z) + d(f(z), y) + d(x, z0) + d(f(z0), y0) + ε} z,z0∈X ≥ inf {d(z, z0) + d(f(z), y) + d(f(z0), y0) + ε} z,z0∈X = inf {d(f(z), f(z0)) + d(f(z), y) + d(f(z0), y0)} z,z0∈X ≥ d(y, y0).

We will use this distance in the following theorem, which shows the usefulness of ε-isometries. This theorem gives a slightly stronger estimate than is given in Corollary 7.3.28 in [9], with a more direct proof.

Proposition 1.2.8. Let X and Y be metric spaces and let ε > 0, then:

3 • if there exists an ε- isometry from X to Y , then dGH (X,Y ) ≤ 2 ε

• if dGH (X,Y ) < ε, then there exists an 2ε- isometry from X to Y . 1.2. THE GROMOV-HAUSDORFF DISTANCE 7

Proof. Suppose f is an ε-isometry. We calculate the Hausdorﬀ distance between X and Y , as subspaces of the disjoint union of X and Y , with the metric d deﬁned prior to the statement of the theorem: ε sup inf d(x, y) = sup inf inf d(x, z) + d(f(z), y) + x∈X y∈Y x∈X y∈Y z∈X 2 ε ≤ sup inf d(x, x) + d(f(x), y) + x∈X y∈Y 2 ε = sup d(f(x), f(x)) + x∈X 2 ε = 2 For the other direction, we use the fact that f(X) is an ε-net. ε sup inf d(x, y) = sup inf inf d(x, z) + d(f(z), y) + y∈Y x∈X y∈Y x∈X z∈X 2 ε ≤ sup inf d(x, x) + d(f(x), y) + y∈Y x∈X 2 ε = sup inf d(f(x), y) + y∈Y x∈X 2 3ε ≤ 2

3 It follows that dGH (X,Y ) ≤ 2 ε. Conversely, suppose that dGH (X,Y ) < ε. Then there exist a metric d on the disjoint union of X and Y such that sup inf d(x, y) < ε and sup inf d(x, y) < ε. y∈Y x∈X x∈X y∈Y Then, with respect to this metric d: [ (B(x, ε) ∩ Y ) = Y x∈X

Furthermore, none of the sets B(x, ε)∩Y are empty, because if one of these sets were empty, it would imply that sup inf d(x, y) ≥ ε. By the axiom of choice, x∈X y∈Y Y (B(x, ε) ∩ Y ) 6= ∅. x∈X Y The claim is that any f ∈ (B(x, ε)∩Y ) is an 2ε- isometry. To see that f(X) x∈X is an ε-net, pick any y in Y , then there exists x ∈ X such that y ∈ B(x, ε), and thus d(f(x), y) < ε. Hence f(X) is an ε- net. To see that the distortion of f is smaller than 2ε, ﬁrst note that for every x, it holds that f(x) ∈ B(x, ε) ∩ Y and thus d(f(x), x) < ε. Using this we can easily see that

d(f(x), f(x0)) ≤ d(f(x), x) + d(x, x0) + d(x0, f(x0)) < d(x, x0) + 2ε and that

d(x, x0) ≤ d(x, f(x)) + d(f(x), f(x0)) + d(f(x0), x0) < d(f(x), f(x0)) + 2ε

These two together imply that the distortion of f is smaller than 2ε. Hence f is an 2ε- isometry between X and Y . 8 CHAPTER 1. CONTEXT

To prove the next theorem, we need a small lemma: Lemma 1.2.9 ([9] Thm.1.6.14). Let h : Y → Y be a distance preserving map. If Y is compact, then h is surjective. Proof. Suppose towards a contradiction that there is a y in Y \ h(Y ). Since Y is compact, h(Y ) is compact as well, and hence there exists a positive ε such that Bε(y) ∩ h(Y ) = ∅. Let S ⊂ Y be a maximal ε- separated set, and denote its cardinality by n. Because h is distance preserving, h(S) is an ε-separated set as well. But then h(S) ∪ {y} is an ε- separated set of cardinality n + 1, contradicting the maximality of S. Theorem 1.2.10 ([9] Thm.7.3.30). Let X and Y be compact metric spaces, then X and Y are isometric if and only if dGH (X,Y ) = 0. Proof. Suppose f : X → Y is an isometry. Then f is an ε- isometry for every 3 positive ε. By proposition 1.2.8, dGH (X,Y ) < 2 ε. Since this holds for all ε, dGH (X,Y ) = 0 Conversely, suppose that dGH (X,Y ) = 0, then for all strictly positive ε, there 1 exists an ε- isometry. Let fn be an n - isometry. By compactness of X, there exists a countable dense subset D = {x1, x2,... } of X. By compactness there exists a converging subsequence of fn(x1), denoted fnk (x1). Repeating this once, we get a subsequence of fn that converges at the points x1 and x2 and repeating this infinitely many times, we get a subsequence of fn that converges on D. Hence we may without loss of generality assume that fn converges pointwise on D. Then we can define for x ∈ D the function f(x) := limn fn(x) and extend 1 this to a continuous function f on the whole of X. Because dist(fn) < n we have that: 1 |d(f (x ), f (x )) − d(x , x )| < n i n j i j n Taking the limit of n to infinity, one sees that f is a distance preserving map, from X to Y . Similarly, one obtains a distance preserving map g from Y to X. Then f ◦g : Y → Y is distance preserving as well and because Y is compact, this implies that f ◦g is surjective. Hence g is surjective and thus is an isometry.

Theorem 1.2.11. Let X and Y be compact metric spaces, then dGH (X,Y ) < ∞. Proof. Define M := sup d(x, x0) and N := sup d(y, y0). By compactness, x,x0∈X y,y0∈Y both N and M are finite. Pick x0 ∈ X and y0 ∈ Y . Define a metric on X t Y by setting: d(y, x) = d(x, y) = d(x, x0) + N + M + d(y0, y), which never attains values greater than 2(N+M). Hence it holds that dH (X,Y ) ≤ 2(N + M) and dGH (X,Y ) < ∞. The conclusion is that the set of all compact, metric spaces modulo isometry and endowed with the Gromov-Hausdorff distance forms a metric space. Chapter 2

Basic Theory

In this chapter, we will review the theory necessary to deﬁne spectral triples and more importantly, the correspondences between them.

2.1 C*-modules

C*-modules are a generalization of Hilbert spaces. The inner product on a C*-module takes its values in a general C*-algebra and a Hilbert space is then obtained as a special case when the C*-algebra is C. A good book containing the theory of C*-modules is the book by Lance [12].

Deﬁnition 2.1.1. Let A be a C*-algebra. A pre-C*-A -module is a right A - module E with a complex vector space structure, equipped with a conjugate, bilinear inner product h·|·i with values in A , such that for all e and f in E and a ∈ A :

1. he|ei ≥ 0

2. he|ei = 0 if and only if e = 0

3. he|fi = hf|ei∗

4. he|fai = he|fi a p A norm on E can be deﬁned by kek := k he|ei k. If E is complete in this norm, it is called a C*-module.

Example 2.1.2. Given a C*-algebra A , the easiest examples of C*-A -modules are the spaces Cn(A ), the direct sum of n copies of A , with A -valued inner n X ∗ product given by h(ai)i|(bi)ii = ai bi. i=1

If H, K are Hilbert spaces, every operator T : H −→ K admits an adjoint operator T ∗ : K −→ H. For C*-modules this does not hold. Therefore we make the following deﬁnitions:

9 10 CHAPTER 2. BASIC THEORY

Deﬁnition 2.1.3. Let E , F be C*-modules. Denote by Hom (E , F ) the Ba- nach space of continuous A -module homomorphisms. We deﬁne

Hom∗ (E , F ) := {T ∈ Hom (E , F ) | ∃T ∗ ∈ Hom (F , E ) hT e|fi = he|T ∗fi ∀e ∈ E , f ∈ F } , the adjointable operators. End(E ) and End∗ (E ) are deﬁned as the continuous, resp. adjointable endomorphisms.

Deﬁnition 2.1.4. Given a C*-A -module E and a C*-B-module F , as well as ∗ a *-homomorphism A −→ EndB (F ), such that the linear span of AE is dense in F , we deﬁne the interior tensor product of E and F over A , denoted by E ⊗b A F , as the completion of E ⊗A F in the norm induced by the following inner product:

he1 ⊗ e2|f1 ⊗ f2i := hf1|he1|e2i f2i

With this norm, E ⊗b A F is a C*-B-module.

Definition 2.1.5. Given a C*-A -module E and two elements x, y ∈ E , we can define the linear operator |xi hy| : E −→ E : z 7→ x hy|zi. Finite linear combinations of these operators are called finite rank operators, denoted by FinA (E ). The closure of FinA (E ) in the operator norm is denoted KA (E ), the A -compact operators on E . The subscript A in the notation is to remind that the operators are right A linear, because:

In the special case where E is a Hilbert space, these definitions coincide with the usual definitions of finite rank and compact operators on a Hilbert space.

Deﬁnition 2.1.6. If A is a C*-algebra, then a contractive approximate identity for A is a net {uλ}λ in A such that kuλk ≤ 1 for all λ and for all a ∈ A it holds that kuλa − ak → 0 and kauλ − ak → 0.

In the following result we will need the following lemma.

Lemma 2.1.7 ([10] 1.7.2). Let A be a C*-algebra, and let I ⊆ A be a dense ideal. Then there is a contractive approximate identity of A , consisting of elements of I. If A is separable, this contractive approximate identity is countable.

Proof. Let Λ be the set of finite subsets of I, ordered by inclusion. For λ = nλ X ∗ {x1, x2, . . . , xnλ } ∈ Λ, define the element vλ = xkxk and define uλ = k=1 1 −1 vλ n + vλ ∈ I. 1 −1 Because the function t 7→ t n + t only takes values between 0 and 1 on the positive real numbers, and because vλ is selfadjoint, we get that 0 ≤ uλ ≤ 1. 2.1. C*-MODULES 11

Observe that

1 −1 uλ − 1 = vλ + vλ − 1 nλ 1 −1 1 1 −1 = vλ + vλ − + vλ + vλ nλ nλ nλ 1 1 −1 = (vλ − − vλ) + vλ nλ nλ 1 1 −1 = − + vλ , nλ nλ and hence that:

nλ nλ X ∗ X ∗ [(uλ − 1)xi][(uλ − 1)xi] = (uλ − 1) xixi (uλ − 1) = (uλ − 1)vλ(uλ − 1) i=1 i=1 1 1 −2 = 2 vλ + vλ . nλ nλ

−2 1 1 1 The function t 7→ 2 t + t only takes values smaller than 2 on the nλ nλ nλ positive real numbers, so for each 0 ≤ i ≤ nλ we get:

nλ ∗ X ∗ [(uλ − 1)xi][(uλ − 1)xi] ≤ [(uλ − 1)xi][(uλ − 1)xi] i=1 1 1 −2 = 2 vλ + vλ nλ nλ 1 ≤ 2 . nλ

1 It follows that k(uλ − 1)xik ≤ 2 and thus that k(uλ − 1)xk → 0 for every x in nλ I. Because I is dense and kuλk ≤ 1 for every λ, we get that k(uλ − 1)ak → 0 for every a ∈ A and hence uλ is an approximate identity for A . If A is separable, there is a dense sequence x1, x2,... in I and then we n X ∗ set vn = xixi and continue the proof in the same way as above to get a i=1 countable approximate identity {un}n.

The following result shows that in a way, all C*-modules are some kind of limit of spaces Cn(A ).

Theorem 2.1.8 (Theorem 3.1 in [2]). Suppose Y is a Banach space that is also a right module over the C*-algebra A . Then Y is a C*-module with norm coinciding with the C*-module norm if and only if there exists a net of positive integers n(α), and contractive module maps φα : Y −→ Cn(α)(A ), ψα : Cn(α)(A ) −→ Y , with ψα ◦ φα → IdY strongly on Y . In this case, the ∗ norm limit limα φα(y) φα(z) exists and is equal to the C*-module inner product. 12 CHAPTER 2. BASIC THEORY

Proof. For a complete proof we refer to [2], but we will sketch what the maps φα and ψα look like when Y is a C*-module. The C*-algebra K(Y ) has FinA (Y ) as a dense ideal, and so using lemma 2.1.7, KA (Y ) admits an approximate identity of the form n(α) X uα = |xki hxk| . k=0 Use this approximate identity to deﬁne:

φα : Y → Cn(α)(A ) ψα : Cn(α)(A ) → Y n(α) n(α) n(α) X y 7→ (hxk|yi)k=0 (ak)k=0 7→ xkak k=0

Another important example of a C*-module is the canonical module HA := H ⊗b A , where H is `2 or any other inﬁnite dimensional separable Hilbert space. It is a theorem of Kasparov, a proof of which can be found in [12], that for any ﬁnitely ∼ generated C*-A -module E, E ⊕ HA = HA .

Deﬁnition 2.1.9. In order to be able to generalize C*-module theory, it is convenient to describe objects metrically, as is done in [2], that is, without using the inner product. Given a C*-A -module E , we can form the left A -module E ∗, which is equal to E as a set, but with the following structure:

∗ ∗ ae := ea he|fiE ∗ = he|fiE

The module E ∗ is called the dual module of E .

Proposition 2.1.10. If E is a C∗-A -module, then the following identity holds:

2.2 Operator spaces

Operator spaces are a generalization of Banach spaces that arise when dealing with noncommutative, also called quantized, mathematics. Loosely speaking, an operator space is a space consisting of operators, but this is made more precise in the next paragraph. A good introduction to the theory of operator spaces is the book by Pisier [14], or the book by Blecher and Le Merdy [3], which more emphasizes operator algebras. The article by Blecher [2] uses operator space theory to describe C*-modules in a metric way.

Deﬁnition 2.2.1. An operator space V is a closed linear subspace of a C*- algebra. 2.2. OPERATOR SPACES 13

According to a theorem of Gelfand and Naimark, for every C*-algebra A , there exist a Hilbert space H and an isometric ∗-isomorphism from A onto a closed ∗-subalgebra of B(H). Thus an operator space V can be regarded as a closed linear subspace of B(H), for some Hilbert space H. The action of V on H then determines an action of the space of n × n-matrices with entries in n V , Mn(V ), on H , given by matrix multiplication. We can use this action to deﬁne a norm k · kn on the space of n × n-matrices with entries in V , denoted by Mn(V ). If A is an n × n-matrix with entries in V , we deﬁne:

kAkn = sup kAhk . h∈Hn Deﬁnition 2.2.2. Given a linear mapping between operator spaces φ : V −→ W , deﬁne

∼ 1⊗φ ∼ φn : Mn(V ) = Mn(C) ⊗ V −→ Mn(C) ⊗ W = Mn(W ). The completely bounded norm of φ is deﬁned:

kφkcb = sup kφnk n∈N

A mapping φ is called a complete isometry if every φn is an isometry and it is called a complete isomorphism if it is a linear isomorphism satisfying −1 kφkcb, kφ kcb < ∞. Remark 2.2.2.1. It is theorem 3.1 in [15] that any Banach space V , equipped with a norm k · kn on each of the spaces Mn(V ), n ∈ N, that satisﬁes

1. kv ⊕ wkm+n = max {kvkm, kwkn} 2. kαvβk ≤ kαk kvk kβk n Mn,m(C) m Mm,n(C) for all v ∈ Mm(V ), w ∈ Mn(V ), α ∈ Mn,m(C) and β ∈ Mm,n(C), is completely isometric to an operator space.

Proposition 2.2.3. If K denotes the C*-algebra of compact operators on the separable Hilbert space `2, and V ⊆ B(H) is an operator space, then the tensor product K ⊗ V is equipped with a norm from its representation on `2 ⊗ H. An equivalent deﬁnition of a completely bounded, respectively a completely contractive map is that the induced map 1⊗φ : K⊗V −→ K⊗W is bounded, respectively contractive. Proof. ( )

X X kφnk = sup Ai ⊗ φ(vi) | Ai ⊗ vi ≤ 1,Ai ∈ Mn(C) i i ( )

X X ≤ sup Ai ⊗ φ(vi) | Ai ⊗ vi ≤ 1,Ai ∈ K i i = k1 ⊗ φk

Since this holds for all n ∈ , k1 ⊗ φk ≥ sup kφ k. N n∈N n 14 CHAPTER 2. BASIC THEORY

m X Conversely, let ε > 0 and pick u = Ai ⊗ vi ∈ K ⊗ V such that kuk = 1 i=0 and k(1 ⊗ φ)(u)k > k1 ⊗ φk − ε. yn yn Since Ai is compact, kAi − Aik → 0 as n → ∞, where A denotes the matrix with the same entries as A in the upper left n × n block, and zeros m X elsewhere. Denote M = kφ(vi)k and let N be such that n > N implies that i=0 yn ε for all i, kAi − Aik < M . Then for such n we get: k1 ⊗ φk − ε < k(1 ⊗ φ)(u)k m m X yn X yn = k Ai ⊗ φ(vi) + (Ai − Ai ) ⊗ φ(vi)k i=0 i=0 m m X yn X yn ≤ kAi ⊗ φ(vi)k + kAi − Ai k kφ(vi)k i=0 i=0 ≤ kφnk + ε. Since this holds for all positive ε, we get that k1 ⊗ φk ≤ sup kφ k. n∈N n The following example is a slight modiﬁcation of section 3 in [1].

p Example 2.2.4. Given p ∈ N>1, z ∈ C , deﬁne:     z1 ··· zp z1 0 ··· 0  0 ··· 0   z2 0 ··· 0  R :=   and C :=   z  . .. .  z  . . .. .   . . .   . . . .  0 ··· 0 zp 0 ··· 0 And deﬁne the two operator spaces:

p p R := {Rz|z ∈ C } and C := {Cz|z ∈ C } When regarded as Banach spaces, these spaces are isometric through φ : R −→ C : Rz 7→ Cz, since:

p 2 ∗ X 2 ∗ 2 kRzk = kRzRzk = |zi| = kCz Czk = kCzk i=1

A short calculation shows that the norm on Mn(C) can be expressed as:   2 | 0 ··· 0 2 C =  . .. .  zij ij  zij . . .  n | 0 ··· 0 ij n   2   | | " n # ∗ X ∗ =  zij  = −zji−  zij  = zkizkj ij | | k=1 ij ij ij

" n # X = hzki|zkji

k=1 ij 2.2. OPERATOR SPACES 15

2 Pp A similar calculation yields for Mn(R) the norm Rzij = [ k=1 hzjk|ziki]ij . ij n p Next, deﬁne for 1 ≤ i, j ≤ p, the elements of C given by zi1 = ei and zij = 0 p for j 6= 1, where ei denotes the standard basis of C . Then we can calculate:

" p # p 2 X X Czij = hzki|zkji = hek|eki ij p k=1 ij k=1 = p Whereas for φn( Czij ) = Rzij , we get that:

" p # 2 X Rzij = hzjk|ziki = [hej|eii]ij ij p k=1 ij = kIdk = 1 And so φ is not a complete isometry. Deﬁnition 2.2.5. If we look at the space C from the last example, we can mimic the expression for the norm obtained there, to deﬁne for any Hilbert space H the following norm on Mn(H):

" n # 2 X k [hij] k := hhki|hkji ij n k=1 ij The space H, made into an operator space with the above norms, is called the Column Hilbert space, usually denoted Hc. It is important to note here that this definition differs from the one in [2]. This is because Blecher uses a inner product that is conjugate linear in the second variable, whereas here all inner products are conjugate linear in the first variable. ∼ Using the identification Mm,n(V ) = Mm,n(C) ⊗ V , it is possible to define the matrix multiplication:

: Mn,r(V ) × Mr,m(W ) −→ Mn,m(V ⊗ W ):(α ⊗ v0, β ⊗ w0) 7→ αβ ⊗ v0 ⊗ w0

And using this multiplication, the Haagerup tensor norm on Mn(V ⊗ W ) is deﬁned by:

kukh := inf {kvkkwk|v ∈ Mn,r(V ), w ∈ Mr,n(W ), u = v w, r ∈ N} For clarity we state that the Haagerup norm on V ⊗ W itself is given by:

kukh = inf {kvkkwk|v ∈ M1,r(V ), w ∈ Mr,1(W ), u = v w, r ∈ N} ( r−1 )

r r X = inf kvkkwk v ∈ V , w ∈ W , u = vi ⊗ wi, r ∈ N i=0 Theorem 2.2.6 ([3] Lemma 3.4.5 and Thm.3.4.10). The completion of V ⊗ W in the Haagerup norm is denoted by V ⊗˜ W . The Haagerup tensor product is associative, and if Ti : Xi → Yi are two completely bounded maps, then T1 ⊗ T2 : X1 ⊗˜ X2 → Y1 ⊗˜ Y2 is completely bounded as well, with the norm satisfying kT1 ⊗ T2kcb ≤ kT1kcbkT2kcb. 16 CHAPTER 2. BASIC THEORY

Deﬁnition 2.2.7. An operator algebra is an operator space A that is also an algebra, such that the multiplication map induces a completely bounded map A ⊗˜ A → A. An operator module M is an operator space that is a module over an operator algebra A, such that the module action induces a completely bounded map M ⊗˜ A A → M. We call an operator module M essential if the linear span of MA is dense in M. Lemma 2.2.8 ([2] Lemma 2.2). Let Y be a Banach space, respectively an operator space, and let {Hα}α be a collection of Hilbert spaces, respectively column Hilbert spaces, together with (completely) contractive maps φα : Y −→ Hα and ψα : Hα −→ Y , such that ψα(φα(y)) → y, for all y ∈ Y . Then Y is a Hilbert space, respectively a column Hilbert space. In this case, the inner product on Y is given by hy|zi = limα hφα(y)|φαzi.

Proof. Since φα, ψα are contractions:

kyk = k lim ψα(φα(y))k = lim kψα(φα(y))k ≤ lim kyk = kyk (2.1) α α α and thus: kyk = lim kφα(y)k α Using the polarization identity, each inner product hφα(x)|φα(y)i can be expressed in terms of the norm of Hα. Combining this with kyk = lim kφα(y)k, α we can see that hx|yi := lim hφα(x)|φα(y)i is well deﬁned and deﬁnes an inner α product. Thus Y is a Hilbert space. If Y is a operator space and φα and ψα are completely contractive, we can modify equation 2.1 to yield k [yij] kn = lim k [φα(yij)] kn. And we can continue: α

1 " n # 2 X k[yij]k = lim hφα(yki)|φα(ykj)i n α k=1 1 " n # 2

X = hyki|ykji k=1

An the last expression is just the norm of [yij] when we regard Y as a Column Hilbert space.

Theorem 2.2.9 ([2] Thm.3.4). If X is a C*-A -module, H a column Hilbert space upon which A is represented, then X⊗˜ A H is a column Hilbert space with inner product given by

hx ⊗ h|y ⊗ ki = hh|hx|yi ki

That is, in this case, the Haagerup tensor product and the C*-module interior tensor product coincide.

Proof. As in the proof of theorem 2.1.8, for every C*-module E , the C*-algebra n(α) X α α K(E ) admits a contractive approximate identity {eα}α of the form eα = |xk i hxk |. k=1 2.2. OPERATOR SPACES 17

Use this approximate identity to deﬁne:

 α  hx|x0 i  .  φα : X −→ Cn(α)(A ): x 7→  .   D E  x xα n(α)   a0 n(α)  .  X α ψα : Cn(α)(A ) −→ X :  .  7→ xk ak k=1 an(α)

Then φα and ψα are completely contractive and ψα ◦ φα converges pointwise to the identity. By theorem 2.2.6, the mappings

φα ⊗ 1I: X ⊗˜ A H −→ Cn(α)(A ) ⊗˜ A H

ψα ⊗ 1I: Cn(α)(A ) ⊗˜ A H −→ X ⊗˜ A H are completely contractive as well and their composition converges pointwise to ∼ the identity. Because Cn(α)(A ) ⊗˜ A H = Cn(α)(H), which is a column Hilbert space, lemma 2.2.8 implies that X ⊗˜ A H is a column Hilbert space with inner product given by:

X α α = h x xk hxk |yi k k and this last expression converges to hh|hx|yi ki with α.

Theorem 2.2.10 (Theorem 3.6 in [4]). If X1 ⊆ X and Y1 ⊆ Y are subspaces of operator spaces, then the inclusion X1 ⊗˜ Y1 ⊆ X ⊗˜ Y is a complete isometry. From this theorem we can also conclude that for elementary tensors x ⊗ y ∈ X ⊗˜ Y , it holds that kx⊗yk = kxkkyk, by considering the space span{x} ⊗˜ span{y}. Given an operator algebra A, the next deﬁnition is made with theorem 2.1.8 in mind. In case A is a C∗-algebra, the deﬁnition below is equivalent to E being a C∗-module. 18 CHAPTER 2. BASIC THEORY

Definition 2.2.11. An operator A-module E is called a countably generated rigged module if there exist completely contractive module maps φn : E −→ HA and ψn : HA −→ E such that ψn ◦ φn converges strongly to the identity on E. It turns out that working with contractive maps is too restrictive, and therefore the following class of objects is introduced in [13]. Definition 2.2.12. An operator A-module E is called a countably generated stably rigged module if there exist completely bounded module maps φ : E −→ HA and ψ : HA −→ E such that ψ ◦ φ equals the identity operator on E. In this case, we define the dual module of E to be:

∗ ∗ ∗ ∗ ∗ E := {e ∈ HomA (E,A)|e ◦ ψn ◦ φn → e } n n where φn : E −→ A and ψn : A −→ E are obtained by composing the n n maps φ and ψ with the projection of HA on A and the inclusion of A in HA, respectively. Note that in the case where E is actually a C*-module, we have that:

n n ! ∗ ∗ X X ∗ e ◦ ψn ◦ φn(e) = e ( xk hxk|ei) = |e (xk)i hxk| (e) k=1 k=1 n ∗ X ∗ and hence that e is the limit of the finite rank operators |e (xk)i hxk| and k=1 thus E∗ = K(E,A). This, together with proposition 2.1.10 motivates us to define the compact operators on a stably rigged operator module E by K(E) := ∗ E ⊗˜ A E . The following lemma is a slight modification of Lemma 3.4.6 in [3].

Lemma 2.2.13. Let E be an essential left A module. Then m : A ⊗˜ A E → E is a completely bounded isomorphism. Proof. From the deﬁnition of a left module, we get that the multiplication map m : A ⊗˜ A E → E is completely bounded. Let uλ be a contractive approximate identity for A. Then the map:

sλ : E → A ⊗˜ A E

e 7→ uλ ⊗ e is completely bounded (in fact it is completely contractive). Because

sλ(m(a ⊗ e)) = uλ ⊗ ae = uλa ⊗ e → a ⊗ e, and because sλ and m are linear and continuous, and elements of the form P ai ⊗ ei are dense in A ⊗˜ A E, it must hold that sλ(m(z)) → z for all z in A ⊗˜ A E. But then:

k[zij]kn = lim k[sλ(m(zij))]kn ≤ ksλkcb · km(zij)k λ And so m−1 is completely bounded as well and m is a completely bounded isomorphism. Because E is essential, the range of m is equal to E. Chapter 3

Connections

3.1 Connections

Connections are used to extend a not necessarily left A-linear operator to a tensor product on the left side over A. The details are as follows. Let A be an operator algebra and let m : A ⊗˜ A → A denote the multiplication map. Define the module of 1-forms: Ω1(A) := ker m. Definition 3.1.1. A derivation δ : A −→ M is a linear map from A into some A-operator module M that satisfies the following Leibniz rule: δ(ab) = δ(a)b + aδ(b). The map d : A → Ω1(A): a 7→ 1 ⊗ a − a ⊗ 1 is a universal derivation in the following sense:

Proposition 3.1.2 ([13] Prop.5.1.2). If δ : A → M is a derivation into an 1 operator A-module M, then there is a unique bimodule map jδ :Ω (A) → M 1 1 such that δ = jδ ◦ d. We deﬁne Ωδ(A) := jδ(Ω (A)).

Proof. Because of the requirement δ = jδ ◦ d we get for elements of the form 1 da ∈ Ω (A) that jδ(da) = δ(a). Because jδ must be a bimodule map, and Ω1(A) is generated as a bimodule by elements of the form da, we get for a 1 P P P general element of Ω (A) that jδ( i aidbici) = i aijδ(dbi)ci = i aiδ(bi)ci. Thus, this jδ is the unique bimodule map satisfying δ = jδ ◦ d. Deﬁnition 3.1.3. Let E be a right A-module and let δ be a derivation. A δ- connection on E is a linear map 1 ∇δ : E → E ⊗˜ A Ωδ(A), satisfying the Leibniz rule

∇δ(ea) = ∇δ(e)a + e ⊗ δ(a).

We call a d-connection a universal connection and denote ∇ = ∇d. If E is a right A-module and ∇ is a universal connection on E, then ∇ can be made into a δ-connection ∇δ by deﬁning ∇δ = (1 ⊗ jδ) ◦ ∇.

19 20 CHAPTER 3. CONNECTIONS

Lemma 3.1.4. When D is an operator on a Hilbert space on which A is represented, the function δ(a) = [D, a] deﬁnes a derivation.

Proof. We check the Leibniz rule:

δ(ab) = [D, ab] = Dab − abD = Dab − aDb + aDb − abD = [D, a] b + a [D, b] = δ(a)b + aδ(b).

For derivations of the form δ(a) = [D, a] as above, we introduce the notations 1 1 ΩD(A) = Ωδ(A) and ∇D = ∇δ = (1 ⊗ jδ) ◦ ∇. Connections can be used to extend an operator that is not necessarily left- A-linear to the tensor product over A, in the following way:

Proposition 3.1.5. Let E and F be a right, resp. left, A-module and let D be a linear operator on F . Let ∇ be a universal connection on E. Then the equation

(1 ⊗∇ D)(e ⊗ f) = ∇D(e)f + e ⊗ Df deﬁnes a well deﬁned operator on E ⊗A F . Proof. We check that the expression yields the same result for ea⊗f and e⊗af:

(1 ⊗∇ D)(ea ⊗ f) = ∇D(ea)f + ea ⊗ Df

= ∇D(e)af + e ⊗ [D, a] f + e ⊗ aDf

= ∇D(e)af + e ⊗ Daf

= (1 ⊗∇ D)(e ⊗ af).

The construction above can be used to deﬁne a universal connection on the tensor product of two modules:

Lemma 3.1.6 ([13] Prop.5.2.1). Let E be a stably rigged B-module, F a stably rigged (B,C)-bimodule and let ∇, ∇0 be universal connections on E and F 0 1 respectively. Then 1 ⊗∇ ∇ : E ⊗˜ B F −→ E ⊗˜ B F ⊗˜ C Ω (C) is a universal connection.

Proof. Because ∇0 is a connection, we get that:

0 0 (1 ⊗∇ ∇ )(e ⊗ fc) = e ⊗ ∇ (fc) + ∇(e)fc = e ⊗ ∇0(f)c + e ⊗ f ⊗ dc + ∇(e)fc 0 = (1 ⊗∇ ∇ )(e ⊗ f)c + e ⊗ f ⊗ dc.

Later, we will need the following lemma: 3.1. CONNECTIONS 21

Lemma 3.1.7 ([13] Thm.5.2.2). Let E be a stably rigged B-module, F a stably rigged (B,C)-bimodule and let ∇, ∇0 be universal connections on E and F respectively. Let D be an completely bounded module map between operator spaces X and Y , then:

0 0 1 ⊗∇ (1 ⊗∇ D) = 1 ⊗1⊗∇∇ D, under the isomorphism: ∼ E ⊗˜ B (F ⊗˜ C X) = (E ⊗˜ B F ) ⊗˜ C X.

Proof. We begin by writing out the deﬁnitions:

0 0 (1 ⊗∇ (1 ⊗∇ D))(e ⊗ f ⊗ x) = e ⊗ (1 ⊗∇ D)(f ⊗ x) + ∇1⊗∇0 D(e)(f ⊗ x) 0 = e ⊗ f ⊗ Dx + e ⊗ ∇D(f)x + ∇1⊗∇0 D(e)(f ⊗ x), and

0 0 (1 ⊗1⊗∇∇ D)(e ⊗ f ⊗ x) = e ⊗ f ⊗ Dx + (1 ⊗∇ ∇ )D(e ⊗ f)x 0 = e ⊗ f ⊗ Dx + e ⊗ ∇ (f)x + ∇ 0 (e)fx. D ∇D

From this we see that we would like to compare the connections ∇1⊗∇0 D = (1 ⊗ j ) ◦ ∇ and ∇ 0 = (1 ⊗ j 0 ) ◦ ∇, and thus the maps j and j 0 . 1⊗∇0 D ∇D ∇D 1⊗∇0 D ∇D These maps are deﬁned by:

0 db 7→ [1 ⊗∇0 D, b] and db 7→ [∇D, b] . We can expand the left-hand side to yield:

[1 ⊗∇0 D, b](f ⊗ x) = (1 ⊗∇0 D)(bf ⊗ x) − b(1 ⊗∇0 D)(f ⊗ x) 0 0 = bf ⊗ Dx + ∇D(bf)x − bf ⊗ Dx − b∇D(f)x 0 0 = ∇D(bf)x − b∇D(f)x 0 = ([∇D, b] f)x.

When the module E is an innerproduct A-module, it is possible to deﬁne a 1 1 pairing E ⊗˜ A Ω (A) × E −→ Ω (A)

(e ⊗ ω, f) := he|fi ω.

1 1 This can in turn be used to deﬁne a pairing E × E ⊗˜ A Ω (A) −→ Ω (A):

(e, f ⊗ ω) := (f ⊗ ω, e)∗.

A *-connection is a connection ∇ for which there exists a connection ∇∗ such that: d he|fi = (e, ∇(f)) − (∇∗(e), f) . If the connection ∇∗ = ∇ satisﬁes this equation, the connection ∇ is said to be Hermitian. At ﬁrst it might not seem clear why this condition would be called Hermitian. To see why, let E be a C*-module and H be a Hilbert space and let D be a 22 CHAPTER 3. CONNECTIONS selfadjoint operator on H and let ∇ be a connection on E. For f,g in E, denote ∇(f) = ef ⊗ ωf and ∇(g) = eg ⊗ ωg. Then if we want to require 1 ⊗∇ D to be selfadjoint, we obtain the equation:

h(1 ⊗∇ D)(f ⊗ h)|g ⊗ ki = hf ⊗ h|(1 ⊗∇ D)(g ⊗ k)i

Denote by ΨE the map implementing the cb-isomorphism:

ΨE : HA ⊗˜ A E → H ⊗˜ E x ⊗ a ⊗ e 7→ x ⊗ ae

Then we deﬁne the Grassmann connection ∼ 1 ∼ 1 d : HA = H ⊗˜ A → H ⊗˜ Ω (A) = HA ⊗˜ A Ω (A)

xi ⊗ a 7→ xi ⊗ da Given an operator module E and an operator S on E, we ﬁnd the following for the operator 1 ⊗d S on HA⊗˜ AE:

ΨE ◦ (1 ⊗d S)(x ⊗ a ⊗ e) = x ⊗ [S, a] e + ΨE(x ⊗ a ⊗ Se) = x ⊗ Sae

So we could say that the operator 1⊗d S is such that ΨE ◦(1⊗d S) = (1⊗S)◦ΨE:

1⊗dS HA ⊗˜ A E −−−−→ HA ⊗˜ A E     ΨE y ΨE y H ⊗˜ E −−−−→1⊗S H ⊗˜ E

Proposition 3.1.9 ([8] Prop.2.6). Let ∇ be a connection on HB. Then there ∗ is a unitary U ∈ B(H ⊕ H) such that PG(1⊗dD) = UPG(1⊗∇D)U . This unitary operator only depends on the connection ∇ and is independent of the operator D.

Proof. Because 1 ⊗∇ D and 1 ⊗d D have the same domain, the image of the projection on the ﬁrst coordinate of G(1 ⊗∇ D) and of G(1 ⊗d D) are the same. Then we can deﬁne an isomorphism G(1 ⊗∇ D) −→ G(1 ⊗d D). If we denote p = PG(1⊗dD) and q = PG(1⊗∇D), then this is given by: 1 0 1 d − ∇ g = q p and g⊥ = vgv∗ = (1 − q) (1 − p). ∇ − d 1 0 1 In a diagram, the map g + g⊥ then looks like this: 3.1. CONNECTIONS 23

 1 0   1 d − ∇  ⊕  ∇ − d 1   0 1  G(1 ⊗d D) ⊕ vG(1 ⊗d D) −−−−−−−−−−−−−−−−−−−−−−−−→ G(1 ⊗∇ D) ⊕ vG(1 ⊗∇ D) x  ∼ ∼ = =y

HB ⊗˜ B H ⊕ HB ⊗˜ B HHB ⊗˜ B H ⊕ HB ⊗˜ B H ∗ − 1 ⊥ ⊥ ⊥∗ 1 ⊥ Next, deﬁne u0 = (gg ) 2 g and u0 = (g g ) 2 g . By construction, these are unitary isomorphisms. Because g : Ran (p) −→ Ran (q) and g⊥ : Ran (1 − p) −→ ⊥ Ran (1 − q), we have that u0 : Ran (p) −→ Ran (q) and u0 : Ran (1 − p) −→ ⊥ Ran (1 − q), and U := u0 ⊕ u0 is a unitary endomorphism of HB ⊗˜ B H. Then we can show that U ∗qU is a projection with Ran (U ∗qU) = Ran (p): x = px + (1 − p)x, Ux = qUx + (1 − q)Ux, x = U ∗Ux = U ∗qUx + U ∗(1 − q)Ux.

∗ ∗ Thus PG(1⊗dD) = p = U qU = UPG(1⊗∇D)U 24 CHAPTER 3. CONNECTIONS Chapter 4

Spectral Triples

4.1 Spectral Triples

Spectral triples encode information about spin manifolds in an algebraic manner. In this thesis I have chosen an axiomatic approach, leaving out the connection with the spin manifolds, but for the reader that is interested, a quick introduction can be found in chapters 3 and 7 of the book by Varilly [16]. G. Cornelissen and B. Mesland are currently writing an article [8] in which they give a gen- eralized concept of a correspondence between spectral triples as introduced by B. Mesland in [13] and they define the length of a correspondence, which enables the definition of a distance between spectral triples as the infimum of the lengths of the correspondences between them. In the next section I will review this theory and end with an example.

Deﬁnition 4.1.1. A spectral triple (A , H, D, π) consists of a C*-algebra A , a faithful representation π : A −→ B(H) and an unbounded selfadjoint operator D, with compact resolvent, such that

A := {a ∈ A | k[D, π(a)]k < ∞} is dense in A . The algebra A is called the Lipschitz algebra of (A , H, D, π). The spectral triple is often denoted by (A, H, D, π), denoting the Lipschitz algebra A instead of the C*-algebra A . The Lipschitz algebra A of a spectral triple can be made into an operator algebra by the representation π:

a 0 π : A −→ M (B(H)) : a 7→ (4.1) 2 [D, a] a Note that for x ∈ H , we have that: a 0 x ax = [D, a] a Dx [D, a]x + aDx ax = Dax and hence π(a) preserves the graph of D.

25 26 CHAPTER 4. SPECTRAL TRIPLES

1 2 1 d Example 4.1.2. The spectral triple of the unit circle is (C(S ),L (S ), i dx , π), where the representation π is given by pointwise multiplication. An application of the chain rule yields that the operator [D, f] is equal to multiplication by df i dx , and so the Lipschitz algebra is just: ( ) df 1 A = f ∈ C(S ) < ∞ dx ess−sup ( ) df 1 = f ∈ C(S ) < ∞ dx sup

(The second equality holds because every f is continuous.) Since ker D = 2 1 ⊥ f ∈ L (S ) f is constant a.e. , and the inverse of D on (ker D) is given by the Volterra operator, which is compact, D has compact resolvent.

4.2 C1-modules

In [13] a Ck-algebra A is defined as a C*-algebra that contains a chain of specific subalgebras Ak ⊆ Ak−1 ⊆ · · · ⊆ A. This chain satisfies the property that k each Ai is dense in Ai+1. Given a C -algebra A and a C*-A-module satisfying certain requirements, a chain of operator submodules Ek ⊆ Ek−1 ⊆ · · · ⊆ E is constructed. Such a module E is then called a CK -module. This is similar to the function spaces of k times continuously differentiable functions Ck(X), and the morphisms between these spaces that respect this differentiability. In this thesis, we will only look at C1-algebras and C1-modules, so we will use the notation A ⊆ A and E ⊆ E to denote the dense subalgebra, resp. submodule of the C*-algebra, resp. C*-module. This is consistent with the notation in earlier chapters where A denoted a C*-algebra, E a C*-module, A denoted an operator algebra and E denoted an operator module. Given a C*-A -module E and a spectral triple (A, H, D, π), the C1-algebra chain is just the Lipschitz algebra in A , A ⊆ A . The Lipschitz algebra A of the spectral triple is made into an operator algebra by the representation π as given in equation 4.1. We can define a C1-structure on E , if E admits X an approximate unit uk = |xii hxi| ∈ FinA (E ), such that the matrices 0<|i|≤k

(hxi|xji)ij are elements of Mn(A) and k hxi|xji kA is uniformly bounded in i and j. We deﬁne maps φ, ψ by:

φ : E −→ HA : e 7→ (hxj|ei)j X ψ : HA −→ E :(aj)j 7→ xjaj j

and we let E := {f ∈ E |φk(f) ∈ HA}. We then equip E with the norm kek = kφ(e)kHA . With these deﬁnitions, E becomes a stably rigged A-module through the maps φ| , ψ| , which by deﬁnition of E automatically have the correct range. E HA We can now state a slight variation on lemma . 4.3. UNBOUNDED SELFADJOINT OPERATORS ON C1-MODULES 27

Lemma 4.2.1. If A ⊆ A is a C1-algebra and K is a Hilbert space on which ∼ A is represented by bounded operators, then B ⊗˜ B K = K unitarily. Proof. When we study the proof of lemma , we see that it suﬃces to show that the multiplication map m : B ⊗˜ B K −→ K is contractive. To show this, we ﬁrst ˜ ∼ remark that because B ⊗B K = B⊗b BK and calculate: kb ⊗ kk2 = |hk|hb|bi ki| = |hk|b∗bki| .

Then we calculate:

km(b ⊗ k)k2 = kbkk2 = hbk|bki = kb ⊗ kk2, and hence, because m is linear, kmk = supkuk=1 km(u)k ≤ 1 and m is contractive.

4.3 Unbounded selfadjoint operators on C1-modules

Most operators in this thesis do not have the nice property of being bounded, however we do require some other properties of operators to be able to build a nice theory. First, unbounded operators are not defined everywhere, but we will require them to be densely defined. Also, we want their graphs to be closed, and call an operator whose graph is closed a closed operator. We introduce the notation for the graph of a unbounded operator G(D) = {(e, De)|e ∈ Dom (D)}. In chapter X of [6], it is proven that if D is a densely defined closed unbounded operator on a Hilbert space, and v denotes the operator E ⊕ E −→ E ⊕ E : ∼ ∗ (e1, e2) 7→ (−e2, e1), then H ⊕ H = G(D) ⊕ vG(D ). In chapter 9 of [12], a requirement is stated for this decomposition to hold for unbounded operators on C*-modules, called regularity. A closed, densely defined selfadjoint operator D on a C*-module is called regular if 1 + D2 has dense range.

Deﬁnition 4.3.1. Let D be an unbounded operator on E. If the closure of G(D) is the graph of a function, then this function is called the closure of D, denoted by D.

Deﬁnition 4.3.2. Let D be a closed unbounded operator on a C1-module E ⊆ E , then a subset X of E is called a core for D if the closure of the restriction of D to X is equal to D.

The theory of unbounded operators on C1-modules was developed in [13] and will be summarized in this paragraph.

Definition 4.3.3. An unbounded densely defined selfadjoint operator D : Dom (D) −→ E, where Dom (D) ⊆ E and E ⊆ E is a C1-module, is called regular if it is closed and the operators (D ± i)−1 are densely defined and have finite norm as operators on E.

Remark 4.3.3.1. In [13], the concept of regularity is also deﬁned for nonselfad- 0 D∗ joint operators D, by using the selfadjoint operator D˜ = . Since all D 0 operators we encounter here will be selfadjoint, we will not discuss this further. 28 CHAPTER 4. SPECTRAL TRIPLES

The proofs from the following proposition and theorem are almost identical to the ones in [13]. Proposition 4.3.4 ([13] Prop.4.5.2). If D is a regular operator, then D2 is densely deﬁned, Dom D2 is a core for D, and the operators D±i : Dom (D) −→ E and 1 + D2 : Dom D2 −→ E are bijections. Proof. We know that (D ± i)−1 has ﬁnite norm and from this we can conclude that these operators extend to bounded operators r± and that for x in Ran (D ± i) we have −1 −1 −1 kDr+xk = kD(D + i) xk ≤ k(D + i)(D + i) xk + k − i(D + i) xk ≤ kxk + k(D + i)−1kkxk and hence that Dr+ has a bounded extension to some operator A. A similar calculation shows the same for Dr−. Because D is closed, we can actually prove that Ran (r+) ⊆ Dom (D) and hence that A equals Dr+, rather than being an extension of it. We do this by picking for any e ∈ E a sequence en in Ran ((D + i)), which is dense in E by assumption, that converges to e. Because r+ and Dr+ are bounded, we then have that r+en → r+e and Dr+en → Dr+e and hence that r+e ∈ Dom (D). Because it also holds that r+(D + i) ⊆ Id, we actually have Ran (r+) = Dom (D). The same kind of reasoning also shows that Ran (r−) = Dom (D). Now let e ∈ Dom (D) and let f ∈ E, then we have that

he|fi = hr±(D ± i)e|fi = he|(D ∓ i)r∓fi and since Dom (D) is dense, we then have (D ∓ i)r∓ = Id and so (D ± i) are surjective operators. Because they also have densely deﬁned inverses, they are injective as well. Now it immediately follows that 1 + D2 = (D + i)(D − i) is bijective as well.

Theorem 4.3.5 ([13] Prop.4.5.4). If E ⊆ E is a C1-module over a C1-algebra B ⊆ B, and D is a selfadjoint densely defined closed operator on E, then D is regular if and only if G(D) ⊕ v(G(D)) =∼ E ⊕ E unitarily. Proof. First, assume that D is regular, then by the previous proposition we have that 1+D2 is bijective as an operator between Dom D2 and E and hence that the operators (1 + D2)−1, D(1 + D2)−1 and D2(1 + D2)−1 are defined on the whole of E. A calculation similar to the one used in the proof of the previous proposition shows that these operators are bounded. Because D is selfadjoint, all of these operators are selfadjoint as well and hence we can define a projection: (1 + D2)−1 D(1 + D2)−1 p = D D(1 + D2)−1 D2(1 + D2)−1

Straightforward calculation shows that Ran (pD) ⊆ G(D) and because (1 + 2 −1 2 2 −1 D ) + D (1 + D ) = 1 we also have that Ran (1 − pD) ⊆ G(D). From the deﬁnition of the map v it is easy to see that G(D) and vG(D) are orthogonal and hence the identity E ⊕ E = G(D) ⊕ vG(D) follows. Conversely, suppose E ⊕ E = G(D) ⊕ vG(D) and let p be the projection on G(D). Because p is a projection p∗ = p and we can write: a b∗ p = , b d 4.4. THE GAP METRIC 29 with a and d selfadjoint. Because p is the projection on G(D), we get that Ran (a) ⊆ Dom (D) and b = Da. Similarly, because Ran (1 − p) ⊆ vG(D), and because a aD 1 − a −aD 1 − p = 1 − = , Da d −Da d we get that Ran (Da) = Ran (−Da) ⊆ Dom (D) and 1 − a = D2a, and thus (1 + D2)a = 1 and so 1 + D2 is surjective. Suppose there exists nonzero e ∈ Dom D2 with (1 + D2)e = 0, then −1 ∈ σ(D2), which is a contradiction because D2 is positive. Hence 1+D2 is injective and thus bijective. Because 1 + D2 = (D + i)(D − i), the operator D + i is surjective and D − i is injective, and because 1 + D2 = (D − i)(D + i), the operators D ± i are bijective. By the inverse mapping theorem (see for instance [6] Thm.III.12.5), the operators (D ± i)−1 are bounded and have ﬁnite norm. Hence D is regular.

4.4 The gap metric

The previous theorem allows us to deﬁne the gap distance between regular operators on C1-modules. The gap distance between operators on Hilbert spaces has been studied in [7] and for operators between C*-modules in [11].

Deﬁnition 4.4.1. The gap metric on the space of unbounded regular, selfadjoint operators is deﬁned by:

0 pgap(D,D ) := PG(D) − PG(D0) where PG(D) denotes the projection on the graph of D. Lemma 4.4.2. If U : H −→ K is a unitary operator between two Hilbert spaces and S, T are two unbounded regular operators on H, then

∗ ∗ dgap(USU ,UTU ) = dgap(S, T ).

Proof. Because

G(USU ∗) = {(k, USU ∗k)|k ∈ Dom (USU ∗)} = {(Uh, USh)|h ∈ Dom (US)} = {(Uh, USh)|h ∈ Dom (S)} = UG(S), we get

∗ ∗ dgap(USU ,UTU ) = PG(USU ∗) − PG(UTU ∗)

= PUG(S) − PUG(T ) ∗ = U(P(G(S) − PG(T ))U

= P(G(S) − PG(T ))

= dgap(S, T ). 30 CHAPTER 4. SPECTRAL TRIPLES

4.5 Morphisms of spectral triples

Deﬁnition 4.5.1. Two spectral triples with common algebra, (A, H1,D1, π1) and (A, H2,D2, π2) are unitarily equivalent if there exists a unitary U : H1 −→ H2 such that:

• π2(a)U = Uπ1(a)

• D2U = UD1 Deﬁnition 4.5.2. A strong Morita equivalence between two spectral triples (A1,H1,D1, π1) and (A2,H2,D2, π2) is given by two triples (Ei, ∇i,Ui), where 1 Ei ⊆ Ei is a C -(Aj,Ai) - bimodule, ∇i is an Hermitian connection and Ui : ˜ Ei ⊗Ai Hi −→ Hj a unitary isomorphism such that:

• Aj = KAi (Ei)

• Ui(ae ⊗ x) = πj(a)Ui(e ⊗ x) a ∈ Aj e ∈ Ei, x ∈ Hi ∗ • Ui DjUi = 1 ⊗∇i Di Example 4.5.3. A unitary equivalence U between two spectral triples is obtained as a Morita equivalence via (A, d, U), (A, d, U ∗). Deﬁnition 4.5.4. A correspondence C = (E, ∇,S,U) from a spectral triple 1 (A1,H1,D1) to another spectral triple (A2,H2,D2) consists of a C -(A2,A1)- bimodule E ⊆ E , together with an Hermitian connection ∇ and a selfadjoint regular unbounded operator S on E with compact resolvent and a unitary oper- ˜ ator U : E⊗A1 H1 −→ H2 that intertwines the left algebra representations and is such that U ◦(S⊗1+1⊗∇ D1) = D2 ◦U and such that [S, ∇] = (S⊗1)◦∇−∇◦S ˜ 1 extends to a bounded operator E → E⊗A1 Ω (A1).

Given two correspondences C1 = (E1, ∇1,S1,U1):(A1,H1,D1, π1) −→ (A2,H2,D2, π2) and C2 = (E2, ∇2,S2,U2):(A2,H2,D2, π2) −→ (A3,H3,D3, π3), these correspondences can be composed to a correspondence C2◦C1 from (A1,H1,D1, π1) to (A3,H3,D3, π3), where the composition is given by: ˜ C2 ◦ C1 = E2⊗A2 E1, 1 ⊗∇2 ∇1,S2 ⊗ 1 + 1 ⊗∇2 S1,U2(1 ⊗ U1) To see that the formulae given for the operator and the connection are correct, we calculate:

∗ ∗ ∗ (1 ⊗ U1) U2 D3U2(1 ⊗ U1) = (1 ⊗ U1) (S2 ⊗ 1 + 1 ⊗∇2 D2) (1 ⊗ U1)

= S2 ⊗ 1 ⊗ 1 + 1 ⊗∇2 (S1 ⊗ 1 + 1 ⊗∇1 D1)

= (S2 ⊗ 1 + 1 ⊗∇2 S1) ⊗ 1 + 1 ⊗∇2 1 ⊗∇1 D1 = (S ⊗ 1 + 1 ⊗ S ) ⊗ 1 + 1 ⊗ D 2 ∇2 1 1⊗∇2 ∇1 1

Deﬁnition 4.5.5. Given a correspondence C = (E, ∇,S,U):(A1,H1,D1, π1) −→ (A2,H2,D2, π2), we deﬁne the following two quantities associated to it:

∗ `gap(C ) = dgap (U D2U, 1 ⊗∇ D1)

= dgap (S ⊗ 1 + 1 ⊗∇ D1, 1 ⊗∇ D1) and ` ( ) = d (B ,B ), H C H A2 KA1 (E) 4.5. MORPHISMS OF SPECTRAL TRIPLES 31 where d denotes the Hausdorﬀ distance, calculated in the space End∗ (E), H A1 and BX denotes the unit ball of the subspace X. We then deﬁne the length of a correspondence C to be:

`(C ) = `gap(C ) + `H (C ). To be able to study the length of a correspondence, we ﬁrst need some lemmata. Lemma 4.5.6 ([8] Cor.2.7). If S, T are two regular unbounded operators on a Hilbert space K, let ∇ be a universal connection on HA, then

dgap(1 ⊗∇ S, 1 ⊗∇ T ) = dgap(S, T ) Proof. First note that by proposition 3.1.9, there is a unitary U such that ∗ ∗ PG(1⊗dS) = UPG(1⊗∇S)U and PG(1⊗dT ) = UPG(1⊗∇T )U . Hence

dgap(1 ⊗∇ S, 1 ⊗∇ T ) = dgap(1 ⊗d S, 1 ⊗d T ).

By lemma 4.2.1, there is a unitary isomorphism HA ⊗˜ A K = H ⊗˜ A ⊗˜ A K −→ H ⊗˜ K. Under this isomorphism, 1 ⊗d S is taken to 1 ⊗ S and 1 ⊗d T is taken to 1 ⊗ T . Hence, PG(1⊗dS) = PG(1⊗S) = 1 ⊗ PG(S) and similarly for T . Then, dgap(1 ⊗d S, 1 ⊗d T ) = 1 ⊗ PG(S) − PG(T ) . It is standard that 1 ⊗ PG(S) − PG(T ) ≤ PG(S) − PG(T ) . For the re- versed inequality, pick a positive ε and let e ∈ E of norm ≤ 1 be such that PG(S) − PG(T ) e ≥ PG(S) − PG(T ) − ε. Let x be an element of H with norm 1. Then: 1 ⊗ PG(S) − PG(T ) ≥ x ⊗ PG(S) − PG(T ) e = kxk PG(S) − PG(T ) e

≥ PG(S) − PG(T ) − ε Since this holds for all ε, we obtain the equality:

dgap(1 ⊗∇ S, 1 ⊗∇ T ) = dgap(S, T )

Theorem 4.5.7 ([8] Cor.2.7). For any connection ∇ on an arbitrary C1 module E, we have: 0 0 dgap(1 ⊗∇ D, 1 ⊗∇ D ) = dgap(D,D ).

Proof. We have just proven this statement for any connection on HA, so all we need to do is transfer a connection from an arbitrary module E to HA. This is ∼ done by stabilizing. It is proven in theorem 4.4.3 in [13], that HA = HA ⊕ E isometrically. We can use this to deﬁne on HA the connection 0 ⊕ ∇. Then we get the following:

0 0 dgap(1 ⊗∇ D, 1 ⊗∇ D ) = dgap(1 ⊗0⊕∇ D, 1 ⊗0⊕∇ D ), and the latter is the gap between operators on HA ⊗ K, which by the previous 0 lemma equals dgap(D,D ). 32 CHAPTER 4. SPECTRAL TRIPLES

Proposition 4.5.8 ([8]). Given a correspondence C1, we have that `(C1) ≥ 0. Proof. This is immediate from the deﬁnition.

Proposition 4.5.9 ([8]). If C1 : S1 → S2 and C2 : S2 → S1 are two correspondences in opposite directions and `(C1) = `(C2) = 0, then (C1, C2) is a strong Morita equivalence.

Proof. The fact that `(C1) = 0 implies that `gap(C1) = 0 and thus: dgap(S1 ⊗

1 + 1 ⊗∇1 D1, 1 ⊗∇1 D1) = 0 and thus G(S1 ⊗ 1 + 1 ⊗∇1 D1) = G(1 ⊗∇1 D1).

From this it follows that Dom (S1 ⊗ 1 + 1 ⊗∇1 D1) = Dom (1 ⊗∇1 D1) and on this domain the operators coincide. Thus S1 ⊗ 1 = 0 and because part of the deﬁnition of a spectral triple is that A1 is represented faithfully on H1, this implies that S1 = 0. A similar calculation shows that S2 = 0 as well. Then it easily seen by comparing the deﬁnitions of Morita equivalence and of correspondences with the fact that `H (C1) = 0 implies A2 = KA1 (E) and that

`H (C2) = 0 implies A1 = KA2 (E) that the pair indeed gives a strong Morita equivalence.

Proposition 4.5.10 ([8]). If C1 : S1 −→ S2 and C2 : S2 −→ S3 are two correspondences, then the length of their composition satisﬁes: `(C1 ◦ C2) ≤ `(C1) + `(C2). Proof. For the part of the length that is deﬁned by the gap metric, denoted `gap, we obtain using lemma 4.5.6: ` ( ◦ ) = d (1 ⊗ U )∗U ∗D U (1 ⊗ U ), 1 ⊗ D gap C2 C1 gap 1 2 3 2 1 1⊗∇2 ∇1 1 ∗ ∗ ∗ ≤ dgap ((1 ⊗ U1) U2 D3U2(1 ⊗ U1), (1 ⊗ U1) (1 ⊗∇2 D2)(1 ⊗ U1)) + d (1 ⊗ U )∗(1 ⊗ D )(1 ⊗ U ), 1 ⊗ D . gap 1 ∇2 2 1 1⊗∇2 ∇1 1

Because 1 ⊗ U1 is unitary, we get by lemma 4.4.2 for the ﬁrst part of this expression: ∗ ∗ ∗ ∗ dgap ((1 ⊗ U1) U2 D3U2(1 ⊗ U1), (1 ⊗ U1) (1 ⊗∇2 D2)(1 ⊗ U1)) = dgap (U2 D3U2, 1 ⊗∇2 D2) = `gap(C2). For the second part we obtain with lemma 3.1.7 and theorem 4.5.7: d (1 ⊗ U )∗(1 ⊗ D )(1 ⊗ U ), 1 ⊗ D = d (1 ⊗ U ∗D U , 1 ⊗ 1 ⊗ D ) gap 1 ∇2 2 1 1⊗∇2 ∇1 1 gap ∇2 1 2 1 ∇2 ∇1 1 ∗ = dgap (U1 D2U1, 1 ⊗∇2 D1) = `gap(C1).

And so we conclude `gap(C2 ◦ C1) ≤ `gap(C2) + `gap(C1). For the part involving Hausdorﬀ distances, denoted `H , we ﬁrst use Theorem ˜ ∼ ˜ ˜ ∗ ∼ 3.3.6 in [13] to get KA1 (E2 ⊗A2 E1) = E2 ⊗A2 KA1 (E1) ⊗A2 E2 and KA2 (E2) = ˜ ∗ ∼ ˜ ˜ ∗ E2 ⊗A2 E2 = E2 ⊗A2 A2 ⊗A2 E2 , both completely isometrically. Then we get: ˜ `H (C2 ◦ C1) = dH (KA1 (E2 ⊗A2 E1),A3) ˜ ≤ dH (A3, KA2 (E2)) + dH (KA2 (E2), KA1 (E2 ⊗A2 E1)) ˜ ˜ ∗ ˜ ˜ ∗ = dH (A3, KA2 (E2)) + dH (E2 ⊗A2 A2 ⊗A2 E2 ,E2 ⊗A2 KA1 (E1) ⊗A2 E2 )

≤ dH (A3, KA2 (E2)) + dH (A2, KA1 (E1)) = `H (C2) + `H (C1). Chapter 5

A correspondence between circles

In this section a correspondence will be given from a circle of given radius to a circle with a radius that is an integer multiple of the ﬁrst. Without loss of generality, the ﬁrst circle is assumed to be the unit circle. 1 2 1 i d Recall that the spectral triple of a circle of radius r is given by (Cr(S ),Lr(S ), r dx ). 1 2 1 Here Cr(S ) and Lr(S ) are parametrised from 0 to 2π and the inner product on L2(S1) is given by: r Z hf|gi = f(t)g(t) rdt

1 The only meaning of the subscript r in the notation Cr(S ) is to remind which spectral triple the algebra C(S1) comes from. 1 Define the C*-module E = C(Sr ) := {f ∈ C([0, 2πr]) | f(0) = f(2πr)} with r−1 1 X C1(S )-valued inner product given by: hf|gi (t) := f(t + k2π)g(t + k2π). k=0 1 1 For g ∈ Cr(S ), h ∈ C1(S ) and f ∈ E , define the left and right algebra action on E by: x (gf)(x) := g f(x) and (fh)(x) := f(x)h(φ (x)) r 1 With these definitions, E is a C*-module. 1 2r−1 To define the C -structure on E , let {xj}j=0 be a set of positive, continuous 2r−1 1 dxj 2 functions on Sr , with kxjk∞, dx < ∞, such that xj is a partition of ∞ j=0 unity subordinate to the set (πj − ε, π(j + 1) + ε). Then for given 0 ≤ j ≤ 2r−1 and t ∈ [0, 2π] there is a unique k such that xj(t + k2π) 6= 0. We can use this partition of unity to make E into a C1-module, with the approximate identity i X given by ui = |xji hxj|. Here we define xj = 0 for j ≥ 2r. j=0 Define the maps:

φ : E −→ HC(S1) : f 7→ (hxj|fi) j∈Z X ψ : HC(S1) −→ E :(aj)j∈Z 7→ xjaj j∈Z

33 34 CHAPTER 5. A CORRESPONDENCE BETWEEN CIRCLES

and let E := f ∈ | φ(f) ∈ H 1 and deﬁne on E the norm kfk = kφ(f)k . E Lip(S ) E HLip(S1) When we expand the formula for the norm on E, we get: 2 2 2 2r−1 kekE = kφ(e)k = (hxj|ei)j=0 C2r (A)

2 1 2 1 U : E⊗˜ Lip(S1)L (S ) −→ Lr(S )

f ⊗ h 7→ (t 7→ f(rt)h(φ1(rt))) is a unitary isomorphism that intertwines the left actions of the algebras. Proof. The map U is well deﬁned on the tensor product over Lip(S1) (by this is meant U(fg ⊗ h) = U(f ⊗ gh)) and intertwines the left action because ∼ hf(rt) = h(t)f(rt). To see that U is unitary, ﬁrst note that E ⊗˜ A H = E ⊗˜ A H completely isometrically. This is through the map e ⊗ h 7→ e ⊗ h. By theorem ˜ ∼ 2.2.9, E ⊗A H = E ⊗b A H. Then we calculate: Z 2π hU(f1 ⊗ h1)|U(f2 ⊗ h2)i = f1(rt)h1(φ1(rt))g2(rt)h2(φ1(rt)) rdt 0 r−1 (k+1) 2π X Z r = f1(rt)h1(rt − k2π)g2(rt)h2(rt − k2π) rdt k2π k=0 r r−1 X Z 2π = f1(x + 2πk)h1(x)f2(x + k2π)h2(x) dx k=0 0 Z 2π = h1(x) hf1|f2i (x) h2(x) dx 0 = hf1 ⊗ h1|f2 ⊗ h2i

X int X int When f = ane r is an element of E , U maps f ⊗ 1 to ane and hence n∈Z n∈Z C(S1) ⊆ Ran (U). But since C(S1) is dense in L2(S1) in the L2-norm and we have already established that U is unitary, it follows that U is surjective and hence U is an unitary isomorphism. Lemma 5.0.12. When we deﬁne a connection ∇ as

2r−1 X ∇(f) := xj ⊗ d hxj|fi , j=0 35

d i d it holds that U ◦ 1 ⊗∇ i dx = r dx ◦ U. Proof. Given f ∈ E, h ∈ L2(S1), it holds that:

2r−1 X d (U(∇i d (f)h))(t) = xj(rt) i , hxj|fi h(φ1(rt)) dx dx j=0 2r−1 r−1 X X df = x (rt) x (φ (rt) + k2π)i (φ (rt) + k2π)h(φ (rt)) j j 1 dx 1 1 j=0 k=0 2r−1 X df = x (rt)x (rt)i (rt)h(φ (rt)) j j dx 1 j=0 df = i (rt)h(φ (rt)) dx 1

X 2 Where the second equality holds because when we diﬀerentiate xj = 1 we j X dxj obtain x = 0, and the third equality holds because for given 0 ≤ j ≤ j dx j 2r − 1 and t in [0, 2π] there is a unique k such that xj(t + k2π) 6= 0. The entire expression then becomes: i d df dh r dx (U(f ⊗ h)) (t) = i dx (rt)h(φ1(rt)) + f(rt)i dx (φ1(rt)) dh = U(∇ d (f)h) + U(f ⊗ i ) i dx dx d = U((1 ⊗∇ i dx )(f ⊗ h))

1 Proposition 5.0.13. The correspondence C = (C(Sr ), 0, ∇,U) is a correspon- 1 2 1 d 1 2 1 i d dence from C1(S ),L1(S ), i dx to Cr(S ),Lr(S ), r dx . Proof. The only things left to check is that the zero operator is regular on E, and that [0, ∇] extends to a bounded operator, both of which follow directly from their deﬁnitions.

Because the operator in the correspondence C is the zero-operator, we get that: d d `gap(C ) = dgap(0 ⊗ 1 + 1 ⊗∇ i , 1 ⊗∇ i ) dx dx d d = d (1 ⊗ i , 1 ⊗ i ) gap ∇ dx ∇ dx = 0 and thus

`(C ) = `gap(C ) + `H (C )

= `H (C )

= d (B ,B 1 ) H K(E) Lipr (S ) ∗ where the Hausdorﬀ distance is calculated in EndA (E). Unfortunately, the norm on E turns out to be very diﬃcult to calculate. We can however, make the following observations: 36 CHAPTER 5. A CORRESPONDENCE BETWEEN CIRCLES

∗ Lemma 5.0.14. Every operator T in EndA (E) is compact.

Proof. For given T ∈ EndLip(S1)(E), and f ∈ E, we have the equality:

r−1  r−1 X X T (f) = T  xj hxj|fi = T (xj) hxj|fi j=0 j=0 and thus: r−1 X T = |T (xj)i hxj| j=0 Hence every endomorphism is compact (even ﬁnite rank).

This means that `( ) = d (B ∗ ,B ). Even though it still seems C H EndA(E) A impossible to calculate this distance, it does seem possible to shed a little bit of ∗ light on the structure of A and of EndA (E), through the following lemma. ∗ Lemma 5.0.15. An operator T ∈ EndA (E) is completely determined by its k k i r t values on the functions exp i r : t 7→ e , for 0 ≤ k < r. Moreover, on each of these functions, the operator T is equal to the left action of a function hk in Lip(S1).

Proof. Because every endomorphism is right-A-linear, for given T ∈ EndLip(S1)(E) X n and f = a exp(i ) ∈ E, it holds that: n r n∈Z r−1 X n X X rn + k T (f) = T ( a exp(i )) = T ( a exp(i )) n r rn+k r n∈Z n∈Z k=0 r−1 X k X = T (exp(i )) a exp(in) r rn+k k=0 n∈Z

k k Since the functions exp(i r ) are nowhere zero, and T (exp i r ) ∈ E, we can write k k 1 T (exp(i r )) = hkexp(i r ) for unique functions hk ∈ Lip(S ). The following lemma can be stated without a proof, as it follows easily. ∗ Lemma 5.0.16. An operator T ∈ EndA (E) belongs to A if and only if all of the functions hk are the same. With these results, it does seem plausible that the value of r affects to what extend A and KA(E) behave differently, and thus that the length of the correspondence C depends on r. Also, in the special case r = 1, corresponding to the identity correspondence, the previous two lemmas indeed show that `(C ) = 0. It might seem more natural, given the way we have just written the right algebra action, to define the C1-structure on E by the maps 1 1 r 1 r 1 φ : C(Sr ) −→ C(S ) ψ : C(S ) −→ C(Sr ) ! r−1 X X X anei n 7→ arn+kein (fk)k 7→ e k fk r i r n∈Z n∈Z k k=0 37

1 where e k fk in the last equation denotes the right action of C(S ) on E. How- i r ever it does not seem to be possible to deﬁne a left C(S1) action on C(S1)r in such a way that φ and ψ become left module maps. When the formulae for φ(f) and φ(hf) are expanded, one concludes that the left action should be t (h · (fk)k )k (t) = h( r )fk(t). However, the right hand side of this equation need not be 2π-periodic. We have now seen that the space of spectral triples can be endowed with a metric, by assigning a length to the correspondences between them. We have also seen one example of a correspondence between two commutative examples of spectral triples, arising from the spin manifolds of the circle. 38 CHAPTER 5. A CORRESPONDENCE BETWEEN CIRCLES Bibliography

[1] W. Arveson. What is an operator space? Notes to a lecture. [2] David P. Blecher. A new approach to Hilbert C*-modules. Mathematische Annalen, 307:253–290, 1997. [3] David P. Blecher and Christian Le Merdy. Operator algebras and their modules—an operator space approach. Number 30 in London Mathematical Society Monographs. Oxford University Press, 2004. [4] David P. Blecher and Vern I. Paulsen. Tensor products of operator spaces. Journal of Functional Analysis, 99:262–292, 1991. [5] A. Connes. Gravity coupled with matter and foundation of noncommutative geometry. Commun. Math. Phys., 182:155–176, 1996. [6] John B. Conway. A Course in Functional Analysis. Number 96 in Graduate Texts in Mathematics. Springer Verlag, 2nd edition, 1990. [7] H.O. Cordes and J.P. Labrousse. The invariance of the index in the metric space of closed operators. J. Math. Mech., 12:693–719, 1963. [8] G. Cornelissen and B. Mesland. Metric structures for spectral triples. [9] Y. Burago D. Burago and S. Ivanov. A Course in Metric Geometry, vol- ume 33 of Graduate Studies in Mathematics. A.M.S., 2001. [10] J. Dixmier. C*-algebras. North-Holland Publishing Company, 1977. [11] K.Shariﬁ. The gap between unbounded regular operators. arXiv:0901.1891 [math.OA]. [12] E.C. Lance. Hilbert C*-modules. Number 210 in London Mathematical Society Lecture note series. Cambridge University Press, 1995. [13] Bram Mesland. Unbounded bivariant K-theory and correspondences in noncommutative geometry. Journal fuer die reine und angewandte Mathe- matik, 2012. arXiv:0904.4383 [math.KT]. [14] G. Pisier. Introduction to Operator Space Theory. Number 294 in Lon- don Mathematical Society Lecture note series. Cambridge University Press, 2003. [15] Zhong-Jin Ruan. Subspaces of C*-algebras. Journal of Functional Analysis, 76:217–230, 1988.

39 40 BIBLIOGRAPHY

[16] J. Varilly. An Introduction to Noncommutative Geometry. European Math- ematical Society, 2006.