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Crossed Products and C*-Dynamics Crossed products and C*-dynamics Eusebio Gardella April 4, 2019 1 Contents Contents i Preface iii Introduction v I Group actions and crossed products 1 1 Introduction 3 2 Some preliminaries 5 2.1 Universal C∗-algebras . 5 2.2 Multiplier algebras . 9 2.3 K-theory . 11 2.4 Hilbert modules and operators . 17 2.5 Morita equivalence . 22 3 Group actions 27 3.1 C∗-dynamical systems . 27 3.2 Topological actions . 28 3.3 Actions on noncommutative C∗-algebras . 29 4 Full and reduced crossed products 35 4.1 Covariant representations and crossed products . 35 4.2 Examples and computations . 42 4.3 Conditional expectations in crossed products by discrete groups 47 5 Duality theory for abelian group actions 51 5.1 The universal property of the full crossed products . 51 5.2 Dual actions and Takai duality . 53 6 Compact group actions 59 6.1 Fixed point algebras . 59 6.2 Eigenspaces of compact groups actions . 62 6.3 Spectra for compact abelian group actions . 65 i ii CONTENTS 7 K-theory of crossed products 73 7.1 Connes' Thom isomorphism . 74 7.2 The Pimsner-Voiculescu exact sequence . 76 7.3 Consequences and applications . 82 II Rokhlin-type properties for finite group actions 85 8 Introduction 87 9 Preliminaries 91 9.1 Strongly self-absorbing C∗-algebras. 91 10 Classification of outer actions on R 93 10.1 Preliminaries on von Neumann algebras . 94 10.2 Outer actions on II1-factors . 100 10.3 Classification of outer actions on R . 102 11 The Rokhlin property 107 11.1 Finite group actions with the Rokhlin property . 107 11.2 Classification of equivariant homomorphisms . 111 11.3 Existence of Rokhlin actions . 118 11.4 Duality for Rokhlin actions . 119 11.5 Crossed products of Rokhlin actions . 119 12 The (weak) tracial Rokhlin property 125 13 Rokhlin dimension 127 14 Strongly outer actions on Z-stable C∗-algebras 129 A Structure and classification of C∗-algebras 131 A.1 The beginnings of the Elliott programme . 131 A.2 Kirchberg algebras: a major success, modulo the UCT . 132 A.3 AH-algebras and the power of Elliott's intertwining argument . 133 A.4 Classification through tracial approximations . 135 A.5 Exotic examples: from Jiang-Su to Villadsen, Rørdam and Toms 135 A.6 The Toms-Winter regularity conjecture . 138 A.7 The end of the story . 142 Bibliography 145 Preface Since the work of Connes in the classification of von Neumann algebras and their automorphisms, group actions have received a great deal of attention. Amenable group actions on the hyperfinite II1-factor were completely classi- fied by Ocneanu, extending earlier results of Connes and Jones. In their work, showing that outer actions have the so-called Rokhlin property was fundamen- tal, as this property allows one to prove classification. For C∗-algebras, the picture is more complicated. For once, it is no longer true that (strong) out- erness implies the Rokhlin property, and there is little hope to classify general group actions unless they have the Rokhlin property. On the other hand, the Rokhlin property is very restrictive, and there are many C*-algebras that do not admit any action with this property. Several weakenings of the Rokhlin property have been introduced to address this problem. Among them, the weak tracial Rokhlin property and Rokhlin dimension (for which Rokhlin di- mension zero is equivalent to the Rokhlin property) have been successfully used to prove structure results for crossed products. Furthermore, actions with these properties seem to be very common. In this course, we will focus on actions of finite groups, and will only oc- casionally comment on actions of more general groups. We will introduce the Rokhlin property, provide many examples, and show that Rokhlin actions can be classified. We will also see that there are natural obstructions to the Rokhlin property, and will present some weaker variants of it: the (weak) tracial Rokhlin property and Rokhlin dimension (with and without commuting towers). These properties are flexible enough to cover many relevant examples, and are strong enough to yield interesting structural properties for their crossed products. Fi- nally, we will prove a recent analog of Ocneanu's theorem for amenable group actions on C∗-algebras, namely, that for actions on classifiable algebras (which are, in particular, Jiang-Su stable), strong outerness is equivalent to the weak tracial Rokhlin property, and also equivalent to finite Rokhlin dimension (in fact, dimension at most one). iii Introduction v Part I Group actions and crossed products 1 Chapter 1 Introduction 3 Chapter 2 Some preliminaries 2.1 Universal C∗-algebras Many of the most relevant C∗-algebras can be expressed as universal C∗- algebras on relatively simple sets of generators and relations. Unlike for groups, universal C∗-algebras do not always exist and there a few subtelties in the the- ory. Here, we review those aspects that will be needed later, and refer the reader to Blackadar's seminal work [4]. Definition 2.1.1. Let G be a set, which we call the set of generators. We do not assume that G be finite, or even countable. We define a relation on G to be an expression of the form ∗ ∗ kp(x1; x1; : : : ; xn; xn)k ≤ r; where p is a polynomial on 2n noncommuting variables, n 2 N, r 2 [0; 1), and x1; : : : ; xn 2 G. Let G be a set of generators and let R be a set of relations. A representation of (G; R) consists of a Hilbert space H and a set fax : x 2 Gg ⊆ B(H) such that kp(a ; a∗ ; : : : ; a ; a∗ )k ≤ r; x1 x1 xn xn ∗ ∗ whenever kp(x1; x1; : : : ; xn; xn)k ≤ r is a relation in R. Definition 2.1.2. We say that the family R of relations on a set G is admissible if there exists a non-zero representation of (G; R), and if there exist constants rx 2 [0; 1), for every x 2 G, such that whenever fax 2 B(H): x 2 Gg is a representation of (G; R), then kaxk ≤ rx for all x 2 G. Universal C∗-algebras defined by admissible relations exist, as we show next. Theorem 2.1.3. Let G be a set of generators, and let R be an admissible set of relations on G. Then there exists a unique C∗-algebra C∗(G : R) containing 5 6 CHAPTER 2. SOME PRELIMINARIES ∗ a generating set fax 2 C (G : R): x 2 Gg satisfying the relations from R, such ∗ that whenever B is another C -algebra containing elements fbx 2 B : x 2 Gg satisfying the relations from R, then there exists a unique homomorphism ∗ ': C (G : R) ! B satisfying '(ax) = bx for all x 2 G. Proof. Denote by A the free ∗-algebra generated by G. Each representation fπx 2 B(H): x 2 Gg of (G; R) induces a ∗-representation π : A!B(H), given by π(x) = πx for all x 2 G. For a 2 A, we define kak = supfkπ(a)k: π is a representation of (G; R)g: Observe that kak ≤ 1 for all a 2 A, because R is admissible. We let C∗(G : R) denote the Hausdorff completion of A with respect to this norm, and note that ∗ ∗ ∗ C (G : R) is a C -algebra. If ax 2 C (G : R) denotes the image of the canonical ∗ generator x 2 A, then it is clear that fax 2 C (G : R): x 2 Gg is a generating set that satisfies the relations from R. ∗ Now let B be another C -algebra containing elements fbx 2 B : x 2 Gg satisfying the relations from R. Then there is a unique ∗-homomorphism '0 : A! B given by '0(x) = bx for all x 2 G. Let σ : B !B(H) be a faithful representation of B on some Hilbert space H. Then σ ◦ '0 is a representation of (G; R), and thus kakC∗(G:R) ≤ kσ('0(a))kB(H) = k'0(a)kB: ∗ It follows that '0 extends uniquely to a homomorphism ': C (G : R) ! B satisfying '(ax) = bx for all x 2 G. Finally, uniqueness of C∗(G : R) follows immediately from its universal property. In the definition of admissible representation, the condition that all gener- ators are uniformly bounded in norm is necessary for a universal C∗-algebra to exist. Example 2.1.4. There is no \universal C∗-algebra generated by a single el- ement". This would correspond to G = fxg and R = ;. The reason is that ∗ if such a C -algebra existed, and ax were the canonical generator in it, the norm of ax would have to be larger than the norm of every element in every C∗-algebra. This is of course not possible, so this algebra does not exist. It is often the case that the relations are described rather informally, par- ticularly when the precise description is clear. We present some examples. Examples 2.1.5. 1. The universal unital C∗-algebra generated by a self- adjoint contraction is the universal C∗-algebra with G = f1; ag and R given by k1a − ak ≤ 0; ka1 − ak ≤ 0; ka∗ − ak ≤ 0; and kak ≤ 1: This C∗-algebra is isomorphic to C([−1; 1]) with the canonical generator being the inclusion of [−1; 1] into C. 2.1. UNIVERSAL C∗-ALGEBRAS 7 2. Similarly, the universal C∗-algebra generated by a self-adjoint contraction is isomorphic to C0([−1; 1] n f0g). 3. The universal C∗-algebra generated by a unitary has G = f1; ug and R given by k1u − uk ≤ 0; ku1 − uk ≤ 0; ku∗u − 1k ≤ 0 and kuu∗ − 1k ≤ 0: This C∗-algebra is isomorphic to C(S1).
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