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Lecture Notes on C -Algebras

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Lecture Notes on C -Algebras Lecture Notes on C∗-algebras Ian F. Putnam January 3, 2019 2 Contents 1 Basics of C∗-algebras 7 1.1 Definition . 7 1.2 Examples . 9 1.3 Spectrum . 11 1.4 Commutative C∗-algebras . 15 1.5 Further consequences of the C∗-condition . 23 1.6 Positivity . 24 1.7 Finite-dimensional C∗-algebras . 28 1.8 Non-unital C∗-algebras . 36 1.9 Ideals and quotients . 40 1.10 Traces . 43 1.11 Representations . 46 1.12 The GNS construction . 52 1.13 von Neumann algebras . 60 2 Group C∗-algebras 65 2.1 Group representations . 65 2.2 Group algebras . 67 2.3 Finite groups . 69 2.4 The C∗-algebra of a discrete group . 71 2.5 Abelian groups . 77 2.6 The infinite dihedral group . 79 2.7 The group F2 ........................... 85 2.8 Locally compact groups . 91 3 Groupoid C∗-algebras 97 3.1 Groupoids . 98 3.2 Topological groupoids . 103 3 4 CONTENTS 3.3 The C∗-algebra of an ´etalegroupoid . 107 3.3.1 The fundamental lemma . 107 3.3.2 The ∗-algebra Cc(G) . 108 3.3.3 The left regular representation . 113 ∗ ∗ 3.3.4 C (G) and Cr (G) . 116 3.4 The structure of groupoid C∗-algebras . 119 3.4.1 The expectation onto C(G0) . 119 3.4.2 Traces on groupoid C∗-algebras . 121 3.4.3 Ideals in groupoid C∗-algebras . 126 3.5 AF-algebras . 135 Preface It is a great pleasure to thank those who have made helpful suggestions and found errors: Christian Skau, Charles Starling and Maria Ramirez-Solano. 5 6 CONTENTS Chapter 1 Basics of C∗-algebras 1.1 Definition We begin with the definition of a C∗-algebra. Definition 1.1.1. A C∗-algebra A is a (non-empty) set with the following algebraic operations: 1. addition, which is commutative and associative 2. multiplication, which is associative 3. multiplication by complex scalars 4. an involution a 7! a∗ (that is, (a∗)∗ = a, for all a in A) Both types of multiplication distribute over addition. For a; b in A, we have (ab)∗ = b∗a∗. The involution is conjugate linear; that is, for a; b in A and λ in C, we have (λa + b)∗ = λa¯ ∗ + b∗. For a; b in A and λ, µ in C, we have λ(ab) = (λa)b = a(λb) and (λµ)a = λ(µa). In addition, A has a norm in which it is a Banach algebra; that is, kλak = jλjkak; ka + bk ≤ kak + kbk; kabk ≤ kakkbk; for all a; b in A and λ in C, and A is complete in the metric d(a; b) = ka−bk. Finally, for all a in A, we have ka∗ak = kak2. 7 8 CHAPTER 1. BASICS OF C∗-ALGEBRAS Very simply, A has an algebraic structure and a topological structure coming from a norm. The condition that A be a Banach algebra expresses a compatibility between these structures. The final condition, usually re- ferred to as the C∗-condition, may seem slightly mysterious, but it is a very strong link between the algebraic and topological structures, as we shall see presently. It is probably also worth mentioning two items which are not axioms. First, the algebra need not have a unit for the multiplication. If it does have a unit, we write it as 1 or 1A and say that A is unital. Secondly, the multiplication is not necessarily commutative. That is, it is not generally the case that ab = ba, for all a; b. The first of these two issues turns out to be a relatively minor one (which will be dealt with in Section 1.8). The latter is essential and, in many ways, is the heart of the subject. One might have expected an axiom stating that the involution is isomet- ric. In fact, it is a simple consequence of the ones given, particularly the C∗-condition. Proposition 1.1.2. If a is an element of a C∗-algebra A, then kak = ka∗k. Proof. As A is a Banach algebra kak2 = ka∗ak ≤ ka∗kkak and so kak ≤ ka∗k. Replacing a with a∗ then yields the result. Next, we introduce some terminology for elements in a C∗-algebra. For a given a in A, the element a∗ is usually called the adjoint of a. The first term in the following definition is then rather obvious. The second is much less so, but is used for historical reasons from operator theory. The remaining terms all have a geometric flavour. If one considers the elements in B(H), operators on a Hilbert space, each of these purely algebraic terms can be given an equivalent formulation in geometric terms of the action of the operator on the Hilbert space. Definition 1.1.3. Let A be a C∗-algebra. 1. An element a is self-adjoint if a∗ = a. 2. An element a is normal if a∗a = aa∗. 3. An element p is a projection if p2 = p = p∗; that is, p is a self-adjoint idempotent. 1.2. EXAMPLES 9 4. Assuming that A is unital, an element u is a unitary if u∗u = 1 = uu∗; that is, u is invertible and u−1 = u∗. 5. Assuming that A is unital, an element u is an isometry if u∗u = 1. 6. An element u is a partial isometry if u∗u is a projection. 7. An element a is positive if it may be written a = b∗b, for some b in A. In this case, we often write a ≥ 0 for brevity. 1.2 Examples Example 1.2.1. C, the complex numbers. More than just an example, it is the prototype. Example 1.2.2. Let H be a complex Hilbert space with inner product denoted < ·; · >. The collection of bounded linear operators on H, denoted by B(H), is a C∗-algebra. The linear structure is clear. The product is by composition of operators. The ∗ operation is the adjoint; for any operator a on H, its adjoint is defined by the equation < a∗ξ; η >=< ξ; aη >, for all ξ and η in H. Finally, the norm is given by kak = supfkaξk j ξ 2 H; kξk ≤ 1g; for any a in B(H). Example 1.2.3. If n is any positive integer, we let Mn(C) denote the set of n×n complex matrices. It is a C∗-algebra using the usual algebraic operations for matrices. The ∗ operation is to take the transpose of the matrix and then take complex conjugates of all its entries. For the norm, we must resort back to the same definition as our last example n kak = supfkaξk2 j ξ 2 C ; kξk2 ≤ 1g; 2 n where k · k2 is the usual ` -norm on C . Of course, this example is a special case of the last using H = Cn, and using a fixed basis to represent linear transformations as matrices. Example 1.2.4. Let X be a compact Hausdorff space and consider C(X) = ff : X ! C j f continuous g: 10 CHAPTER 1. BASICS OF C∗-ALGEBRAS The algebraic operations of addition, scalar multiplication and multiplication are all point-wise. The ∗ is point-wise complex conjugation. The norm is the usual supremum norm kfk = supfjf(x)j j x 2 Xg for any f in C(X). This particular examples has the two additional features that C(X) is both unital and commutative. Extending this slightly, let X be a locally compact Hausdorff space and consider C0(X) = ff : X ! C j f continuous, vanishing at infinity g: Recall that a function f is said to vanish at infinity if, for every > 0, there is a compact set K such that jf(x)j < , for all x in X n K. The algebraic operations and the norm are done in exactly the same way as the case above. This example is also commutative, but is unital if and only if X is compact (in which case it is the same as C(X)). Example 1.2.5. Suppose that AB are C∗-algebras, we form their direct sum A ⊕ B = f(a; b) j a 2 A; b 2 Bg: The algebraic operations are all performed coordinate-wise and the norm is given by k(a; b)k = maxfkak; kbkg; for any a in A and b in B. There is an obvious extension of this notion to finite direct sums. Also, ∗ if An; n ≥ 1 is a sequence of C -algebras, their direct sum is defined as 1 ⊕n=1An = f(a1; a2;:::) j an 2 An; for all n; lim kank = 0g: n Aside from noting the condition above on the norms, there is not much else to add. Exercise 1.2.1. Let A = C2 and consider it as a ∗-algebra with coordinate- wise addition, multiplication and conjugation. (In other words, it is C(f1; 2g).) 1. Prove that with the norm k(α1; α2)k = jα1j + jα2j A is not a C∗-algebra. 1.3. SPECTRUM 11 2. Prove that k(α1; α2)k = maxfjα1j; jα2jg is the only norm which makes A into a C∗-algebra. (Hint: Proceed as follows. First prove that, in any C∗-algebra norm, (1; 0); (1; 1); (0; 1) all have norm one. Secondly, show that for any (α1; α2) in A, there is a unitary u such that u(α1; α2) = (jα1j; jα2j). From this, deduce that k(α1; α2)k = k(jα1j; jα2j)k. Next, show that if a; b have norm one and 0 ≤ t ≤ 1, then kta + (1 − t)bk ≤ 1.
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