The Second Summer School on Operator Algebras and Noncommutative Geometry 2016 East China Normal University, Shanghai Lecture 1: Group C*-algebras and Actions of Finite 11–29 July 2016 Groups on C*-Algebras Lecture 1 (11 July 2016): Group C*-algebras and Actions of Finite Groups on C*-Algebras N. Christopher Phillips Lecture 2 (13 July 2016): Introduction to Crossed Products and More Examples of Actions. University of Oregon Lecture 3 (15 July 2016): Crossed Products by Finite Groups; the 11 July 2016 Rokhlin Property. Lecture 4 (18 July 2016): Crossed Products by Actions with the Rokhlin Property. Lecture 5 (19 July 2016): Crossed Products of Tracially AF Algebras by Actions with the Tracial Rokhlin Property. Lecture 6 (20 July 2016): Applications and Problems. N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 1 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 2 / 28 A rough outline of all six lectures General motivation The material to be described is part of the structure and classification The beginning: The C*-algebra of a group. theory for simple nuclear C*-algebras (the Elliott program). More Actions of finite groups on C*-algebras and examples. specifically, it is about proving that C*-algebras which appear in other parts of the theory (in these lectures, certain kinds of crossed product Crossed products by actions of finite groups: elementary theory. C*-algebras) satisfy the hypotheses of known classification theorems. More examples of actions. Crossed products by actions of finite groups: Some examples. To keep things from being too complicated, we will consider crossed products by actions of finite groups. Nevertheless, even in this case, one The Rokhlin property for actions of finite groups. can see some of the techniques which are important in more general cases. Examples of actions with the Rokhlin property. Crossed products of AF algebras by actions with the Rokhlin property. Crossed product C*-algebras have long been important in operator algebras, for reasons having nothing to do with the Elliott program. It has Other crossed products by actions with the Rokhlin property. generally been difficult to prove that crossed products are classifiable, and The tracial Rokhlin property for actions of finite groups. there are really only three cases in which there is a somewhat satisfactory Examples of actions with the tracial Rokhlin property. theory: actions of finite groups on simple C*-algebras, free minimal actions Crossed products by actions with the tracial Rokhlin property. of groups which are not too complicated (mostly, not too far from Z d ) on Applications of the tracial Rokhlin property. compact metric spaces, and “strongly outer” actions of such groups on simple C*-algebras. N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 3 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 4 / 28
Background Do the exercises! These lectures assume some familiarity with the basic theory of C*-algebras, as found, for example, in Murphy’s book. K-theory will be There will be many exercises given, with varying levels of difficulty. To occasionally used, but not in an essential way. A few other concepts will really get to understand this material, please do them! be important, such as tracial rank zero. They will be defined as needed, and some basic properties mentioned, usually without proof. Various side comments will assume more background, but these can be skipped. (Many I am happy to talk to people about the exercises. side comments which should be made will be omitted entirely, for lack of time.) N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 5 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 6 / 28 The beginning: The group ring The beginning: The group ring (continued) Let G be a finite group. Its group ring C [ G ] (a standard construction in G is a discrete group, and � � � algebra) is, as a vector space, the set of formal linear combinations � C [ G ] = a g · g : a g ∈ C , a g = 0 for all but finitely many g ∈ G . a g · g (1) g ∈ G g ∈ G Multiplication: ( a · g )( b · h ) = ( ab ) · ( gh ), extended linearly. of group elements with coefficients a g ∈ C . (Formally: the free C -module Recall the usual polynomial ring C [ x ]. Let � S � denote the ideal generated on G .) The multiplication is ( a · g )( b · h ) = ( ab ) · ( gh ) for g , h ∈ G and by a set S . Also, abbreviate Z / n Z to Z n . (We won’t use p -adic integers.) a , b ∈ C , extended linearly. That is, the product comes from the group. Example G need not be finite (but must be discrete), provided that in (1) one uses = C [ x ] / � x 2 − 1 � . The identity of the group is 1 Take G = Z 2 . Then C [ G ] ∼ only finite sums ( a g = 0 for all but finitely many g ∈ G ). and the nontrivial element is x . Exercise Prove that C [ G ] is an associative unital algebra over C . Exercise (easy) Check the statements made in the previous example. One can use any field (even ring) in place of C . The algebraists actually do this. Exercise (easy) Motivation: Representation theory (brief comments below). Generalize the previous example to Z n for n ∈ N . N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 7 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 8 / 28
The beginning: The group ring (continued) The beginning: The C*-algebra of a group G is a discrete group, and G is a discrete group, and � � � � � � C [ G ] = a g · g : a g ∈ C , a g = 0 for all but finitely many g ∈ G . C [ G ] = a g · g : a g ∈ C , a g = 0 for all but finitely many g ∈ G . g ∈ G g ∈ G Multiplication: ( a · g )( b · h ) = ( ab ) · ( gh ), extended linearly. Make C [ G ] a *-algebra by making the group elements unitary: g ∗ = g − 1 . Multiplication: ( a · g )( b · h ) = ( ab ) · ( gh ), extended linearly. C*-algebraists thus usually write u g for the element g ∈ C [ G ]. So = C [ x ] / � x 2 − 1 � , with the nontrivial group element being x . Recall: C [ Z 2 ] ∼ � � � C [ G ] = a g · u g : a g ∈ C , a g = 0 for all but finitely many g ∈ G . Example g ∈ G Take G = Z . Then C [ G ] ∼ = C [ x , x − 1 ], the ring of Laurent polynomials in The multiplication is ( a · u g )( b · u h ) = ( ab ) · u gh for g , h ∈ G and a , b ∈ C , one variable. The group element k ∈ Z is x k . extended linearly, and the adjoint is ( a · u g ) ∗ = au g − 1 . Exercise Exercise (easy) Show that this adjoint in C [ G ] is well defined, conjugate linear, reverses Check the statements made in the previous example. multiplication: ( xy ) ∗ = y ∗ x ∗ , and satisfies ( x ∗ ) ∗ = x . N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 9 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 10 / 28 The beginning: The C*-algebra of a group (continued) The beginning: The C*-algebra of a group (continued) G is a discrete group, and G is a discrete group. Multiplication in C [ G ]: ( a · u g )( b · u h ) = ( ab ) · u gh . � � � Adjoint: ( a · u g ) ∗ = au g − 1 . C [ G ] = a g · u g : a g ∈ C , a g = 0 for all but finitely many g ∈ G . If w is a unitary representation of G on a Hilbert space H (a group g ∈ G Multiplication: ( a · u g )( b · u h ) = ( ab ) · u gh . Adjoint: ( a · u g ) ∗ = au g − 1 . homomorphism g �→ w g from G to the group U ( H ) of unitary operators on H ), then the unital *-homomorphism π w : C [ G ] → L ( H ) is We still need a norm. We will assume G is finite; things are otherwise � � � � more complicated. First, a bit of representation theory. π w a g · u g = a g · w g . Let G be a discrete group, and let H be a Hilbert space. Let w be a g ∈ G g ∈ G unitary representation of G on H : a group homomorphism g �→ w g from G to the group U ( H ) of unitary operators on H . Then we define a linear Theorem map π w : C [ G ] → L ( H ) by The assignment w �→ π w is a bijection from unitary representations of G � � � � on H to unital *-homomorphisms C [ G ] → L ( H ). π w a g · u g = a g · w g . g ∈ G g ∈ G Exercise Exercise Prove this theorem. (To recover w from π w , look at π w ( u g ).) Prove that π w is a unital *-homomorphism. N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 11 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 12 / 28
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