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 noncom- mutative 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 work- ing 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 cor- respondence ([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 def- inition 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 2 A is such that a = a∗, then jjajj = r(a), where r(a) denotes the spectral radius of a. Proof. By the axioms of a C*-algebra, jjajj2 = jja∗ajj = jja2jj. By induction n 1 1 2 n n it holds for each n ≥ 1, that jja jj 2 = jjajj. Then r(a) = limn jja jj n = 2k 1 limk jja jj 2k = jjajj. Lemma 1.1.2 ([6] Thm.VII.8.1). If A is a Banach algebra that is also a divi- sion ring, then A =∼ C. Proof. For a 2 A , denote by σ(a) the spectrum of a, which is nonempty. For λ 2 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 2 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 2 I, so π(a) 2 π(M) = f0g. So A =M is a field.