Lecture 2: Introduction to Crossed Products and More 1129 July 2016 - PowerPoint PPT Presentation

The Second Summer School on Operator Algebras and Noncommutative Geometry 2016 East China Normal University, Shanghai Lecture 2: Introduction to Crossed Products and More 11–29 July 2016 Examples of Actions 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 University of Oregon Examples of Actions. Lecture 3 (15 July 2016): Crossed Products by Finite Groups; the 13 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) Crossed Products 13 July 2016 1 / 28 N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 2 / 28 A rough outline of all six lectures Recall: Group actions on C*-algebras Definition The beginning: The C*-algebra of a group. Let G be a group and let A be a C*-algebra. An action of G on A is a Actions of finite groups on C*-algebras and examples. homomorphism g �→ α g from G to Aut( A ). Crossed products by actions of finite groups: elementary theory. More examples of actions. That is, for each g ∈ G , we have an automorphism α g : A → A , and Crossed products by actions of finite groups: Some examples. α 1 = id A and α g ◦ α h = α gh for g , h ∈ G . The Rokhlin property for actions of finite groups. We saw some examples coming from actions on compact Hausdorff spaces. Examples of actions with the Rokhlin property. We also saw the inner action: if g �→ z g is a (continuous) homomorphism Crossed products of AF algebras by actions with the Rokhlin property. from G to the unitary group U ( A ) of a unital C*-algebra A , then Other crossed products by actions with the Rokhlin property. α g ( a ) = z g az ∗ g defines an action of G on A . (We write α g = Ad( z g ).) The tracial Rokhlin property for actions of finite groups. Exercise Examples of actions with the tracial Rokhlin property. Prove that g �→ Ad( z g ) really is a continuous action of G on A . Crossed products by actions with the tracial Rokhlin property. Finally, we looked at one example of an infinite tensor product action Applications of the tracial Rokhlin property. (next slide). N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 3 / 28 N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 4 / 28

An infinite tensor product action Infinite tensor product action example (continued) Let A n = ( M 2 ) ⊗ n , the tensor product of n copies of the algebra M 2 of Recall: A n = ( M 2 ) ⊗ n ∼ = M 2 n . 2 × 2 matrices. Thus A n ∼ = M 2 n . Define ϕ n : A n → A n +1 = A n ⊗ M 2 is ϕ n ( a ) = a ⊗ 1, and A = lim → n A n . − ϕ n : A n → A n +1 = A n ⊗ M 2 � 1 0 � ∈ U ( M 2 ), and z n = v ⊗ n ∈ U ( A n ). v = 0 − 1 by ϕ n ( a ) = a ⊗ 1. Let A be the (completed) direct limit lim → n A n . (This is − just the 2 ∞ UHF algebra.) Define a unitary v ∈ M 2 by α n ∈ Aut( A n ) is α n = Ad( z n ). � 1 � Commutative diagram to define the order 2 automorphism α ∈ Aut( A ): 0 v = . 0 − 1 ϕ 0 ϕ 1 ϕ 2 ϕ 3 − − − − → M 2 − − − − → M 4 − − − − → M 8 − − − − → · · · − − − − → A C      Define z n ∈ A n by z n = v ⊗ n . Define α n ∈ Aut( A n ) by α n = Ad( z n ), that      � α 0 � α 1 � α 2 � α 3 � α is, α n ( a ) = z n az ∗ n for a ∈ A . Then α n is an inner automorphism of order 2. ϕ 0 ϕ 1 ϕ 2 ϕ 3 Using z n +1 = z n ⊗ v , one can easily check that ϕ n ◦ α n = α n +1 ◦ ϕ n for C − − − − → M 2 − − − − → M 4 − − − − → M 8 − − − − → · · · − − − − → A all n (diagram on next slide). and it follows that the α n determine an The action of Z 2 is not inner (see later), although it is “approximately order 2 automorphism α of A . Thus, we have an action of Z 2 on A . inner” (that is, a pointwise limit of inner actions). Exercise: Prove that ϕ n ◦ α n = α n +1 ◦ ϕ n . N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 5 / 28 N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 6 / 28 General infinite tensor product actions More examples of product type actions → n ( M 2 ) ⊗ n with the action of Z 2 generated by the direct We had A = lim − limit automorphism � 1 � ⊗ n 0 We will later use the following two additional examples: lim Ad − → 0 − 1 n   1 0 0 We write this automorphism as � ∞ � ∞   Ad 0 1 0 on A = M 3 , � 1 � � ∞ � ∞ 0 0 0 − 1 n =1 n =1 Ad on A = M 2 . 0 − 1 n =1 n =1 and � ∞ � ∞ � � In general, one can use an arbitrary group, one need not choose the same Ad diag( − 1 , 1 , 1 , . . . , 1) on A = M 2 n +1 . unitary representation in each tensor factor (indeed, the actions on the n =1 n =1 factors need not even be inner), and the tensor factors need not all be the In the second one, there are supposed to be 2 n ones on the diagonal, same size (indeed, they can be arbitrary unital C*-algebras, provided one is giving a (2 n + 1) × (2 n + 1) matrix. careful with tensor products). If all the factors are finite dimensional matrix algebras (not necessarily of the same size) and the action in each factor is inner, the action is frequently called a “product type action”. N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 7 / 28 N. C. Phillips (U of Oregon) Crossed Products 13 July 2016 8 / 28