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An Introduction to Noncommutative Topology Francesca Arici

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An Introduction to Noncommutative Topology Francesca Arici An introduction to noncommutative topology Francesca Arici Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, the Netherlands E-mail address: [email protected] This work was partially funded by the Netherlands Organisation of Scientific Research (NWO) under the VENI grant 016.192.237. Abstract. C∗{algebras provide an elegant setting for many problems in mathematics and physics. In view of Gelfand duality, their study is often referred to as noncommutative topology: general noncommutative C∗{algebras are interpreted as noncommutative spaces. These form an established research field within mathematics, with applications to quantum theory and other areas where deformations play a role. Many classical geometric and topological concepts can be translated into operator algebraic terms, leading to the so-called noncommutative geometry (NCG) dictionary. These lectures aim to provide the participants with the tools to understand, consult and use the NCG dictionary, covering the basic theory of C∗-algebra and their modules. Focus will be given on examples, especially those that come from deformation theory and quantisation. Contents Chapter 1. An introduction to commutative C∗-algebras5 Motivation 5 1. Trading spaces for algebras5 Chapter 2. Noncommutative C∗-algebas, representations, and the GNS construction 19 1. Representation of C∗-algebras 19 2. States and the GNS construction 22 3. Notable C∗-algebras 25 Chapter 3. Modules as bundles 29 1. Modules and fiber bundles 29 2. Line bundles, Self-Morita equivalence bimodules, and the Picard group 34 Chapter 4. Cuntz{Pimsner algebras and circle bundles 37 1. Pimsner algebras 37 2. Circle actions 39 Bibliography 41 3 CHAPTER 1 An introduction to commutative C∗-algebras Motivation Over the past decades, the term noncommutative topology has come to indicate the study of C∗-algebras. In this lecture series, I have tried to stay as closed as possible to the original moti- vation: I will make use of many results and definition stemming from the theory of C∗-algebras, the focus will always be on the topological aspects. In the spirit of Connes' noncommutative geometry, I will think of a C∗-algebra as some sort of generalised topological space. The goal of this lecture series is to provide the reader with some basic tools to understand the so-called NCG dictionary, or at least its topological entries: operator algebra topology commutative C* algebra locally compact Hausdorff space unitality compactness * homomorphisms continuous proper functions * automorphisms homeomorphisms separability metrizability finitely generated projective module vector bundle equivalence bimodule line bundle f.g.p. C∗-module Hermitian vector bundle self-Morita equivalence Hermitian line bundle quantum group coaction Lie group action Hopf{Galois extension principal bundle 1. Trading spaces for algebras Let X be a compact, Hausdorff topological space. The set of continuous functions X ! C (with respect to the standard topology on the set of complex numbers C) will be denoted by C(X). We are interested in the properties of this set, which, as we will see, gives a complete algebraic characterisation of the underlying topological space. (1) It inherits a complex vector space structure from C: for all f; g 2 C(X), and for every λ 2 C we can define the function f + λg as (f + λg)(x) = f(x) + λg(x): (2) It is a commutative, unital algebra: 5 (a) multiplication between functions is defined point-wise, namely (fg)(x) := f(x)g(x) for all x 2 X, and this defines a commutative product, namely fg = gf for all f; g 2 C(X); (b) the constant function 1(x) ≡ 1 is the unit element for this multiplication; (c) the product is compatible with the vector space structure; (3) There is a natural norm on the algebra C(X) (1) kfk := sup jf(x)j; x2X and the norm is compatible with the algebra structure, i.e. kfgk ≤ kfk kgk. This makes (C(X); k · k) into a normed algebra. (4) C(X) is complete with respect to the norm defined in (1), i.e. every Cauchy sequence in C(X) converges in the topology defined by the norm (1). One says that (C(X); k·k) is a Banach algebra. (5) The Banach algebra C(X) possesses a ∗-structure, namely an antilinear map ∗ : C(X) ! C(X) such (f ∗)∗ = f (∗ is thus called an antilinear involution): this is defined by ∗ f (x) := f(x), where z is the complex conjugate of z 2 C. It is compatible with the ∗-structure: kf ∗k = kfk. (6) The norm satisfies the C∗-property, namely kf ∗fk = kfk2 for all f 2 C(X): Indeed kf ∗fk = sup jf(x)f(x)j = sup jf(x)j2 = kfk2: x2X x2X The above list of properties can be summarized by saying that C(X) is a commutative unital C∗-algebra Definition 1.1. An (abstract) C∗-algebra is a Banach algebra (A; k · k) with the property that (2) ka∗ak = kak2 for every a 2 A. The C∗-property (2) is crucial, as it ties together analytical and algebraic properties and has profound implications, giving C∗-algebras a very special place within the zoo of Banach algebras. The most striking one, perhaps, is that, as we will see, the norm of a (commutative) C∗-algebra can be given in purely algebraic terms. Our goal is to prove that, actually, all commutative C∗-algebras are of the form C(X) for some compact Hausdorff space X: Theorem 1.1 (Gel'fand{Na˘ımark). Let A be a commutative, unital C∗-algebra. Then there exists a compact Hausdorff space X = σ(A), called the spectrum of the algebra A, such that A is isometrically ∗-isomorphic to the algebra C(σ(A)) of continuous functions on σ(A). Exercise 1.1. Show that the same definitions as above make the space Cb(X) := f 2 C(X) sup jf(x)j < 1 x2X of bounded continuous functions on X into a unital, commutative C∗-algebra. 1.1. Spectral theory for Banach algebras. In this Section, we recall some basic facts from the spectral theory of Banach algebras. 6 1.1.1. Invertibility and the spectrum. Recall that an element a of a unital algebra (or ring) A is called invertible if there is an element b 2 A such that ab = ba = 1. Exercise 1.2. Let A be a unital ring. (1) Prove that if a 2 A is invertible, then there is a unique b 2 A satisfying ab = ba = 1. This b is usually denoted a−1. (2) Given a 2 A, suppose that there exist elements b; c 2 A such that ab = ca = 1. Prove that b = c (and hence that a is invertible). (3) Show that if a and b are invertible, then ab is invertible with (ab)−1 = b−1a−1. (So the set of invertible elements of A forms a group under multiplication.) (4) Show that if a and b commute (i.e., ab = ba), then ab is invertible if and only if both a and b are invertible. (5) Find an example of a unital C∗-algebra A and elements a; b 2 A such that ab = 1 but a and b are not invertible. (6) Now suppose that A is a C∗-algebra, and that a 2 A is invertible. Show that a∗ is invertible, with (a∗)−1 = (a−1)∗. Definition 1.2. Let A be a Banach algebra with unit element 1. The spectrum of an element a 2 A is the set σ(a) := fz 2 C j z1 − a is not invertible in Ag: The resolvent of a is the complement of the spectrum: C n σ(a) = fz 2 C j z1 − a is invertible in Ag; and the resolvent function for a is the function −1 Ra : C n σ(a) ! A; Ra(z) := (z1 − a) : The notion of spectrum generalises both the range of a function, and the set of eigenvalues of a matrix. Exercise 1.3. Consider the C∗-algebra C(X) of continuous functions on a compact Haus- dorff space X. Prove that for each f 2 C(X) one has σ(f) = range(f) = ff(x) 2 C j x 2 Xg: ∗ Exercise 1.4. Consider the C -algebra Mn(C) of n × n matrices. Prove that for each a 2 Mn(C) one has σ(a) = feigenvalues of ag: Exercise 1.5. Show that in a unital C∗-algebra one has σ(a∗) = σ(a) = fz j z 2 σ(a)g. Exercise 1.6 (Properties of the Spectrum). Let A be a unital Banach algebra. (1) Show that for all a; b 2 A one has σ(ab) [ f0g = σ(ba) [ f0g: Hint: prove that if z1 − ba is invertible, then (1 + a(z1 − ba)−1b)(z1 − ab) = z1. (2) Find an example where σ(ab) 6= σ(ba). (3) Show that for each w 2 C and each a 2 A one has σ(w1 − a) = w − σ(a), where the right-hand side is defined to be the set fw − z j z 2 σ(a)g. 7 Pn i (4) Show that for each polynomial p = i=0 pix 2 C[x] and each a 2 A one has σ(p(a)) = p(σ(a)) Pn i where we define p(a) = i=0 pia , and p(σ(a)) = fp(z) 2 C j z 2 σ(a)g. This equality is (an example of) the spectral mapping property.( Hint: write the polynomial z1 − p as a product of linear factors.) Exercise 1.7. Let φ : A ! B be a homomorphism of unital Banach algebras (i.e., a linear map satisfying φ(a1a2) = φ(a1)φ(a2) for all a1; a2 2 A, and φ(1A) = 1B).
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