Lecture 2: Crossed Products by Finite Groups; the Rokhlin Property - PowerPoint PPT Presentation

2014 Summer School for Operator Algebras East China Normal University, Shanghai Lecture 2: Crossed Products by Finite Groups; the Rokhlin Property 26–31 July 2014 N. Christopher Phillips Lecture 1 (26 July 2014): Actions of Finite Groups on C*-Algebras and Introduction to Crossed Products. University of Oregon Lecture 2 (27 July 2014): Crossed Products by Finite Groups; the Rokhlin Property. 27 July 2014 Lecture 3 (28 July 2014): Crossed Products by Actions with the Rokhlin Property. Lecture 4 (29 July 2014): Crossed Products of Tracially AF Algebras by Actions with the Tracial Rokhlin Property. Lecture 5 (30 July 2014): Examples and Applications. N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 1 / 29 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 2 / 29 A rough outline of all five lectures Crossed products by finite groups Let α : G → Aut( A ) be an action of a finite group G on a C*-algebra A . Actions of finite groups on C*-algebras and examples. As a vector space, C ∗ ( G , A , α ) is the group ring A [ G ], consisting of all Crossed products by actions of finite groups: elementary theory. finite formal linear combinations of elements in G with coefficients in A . We conventionally write u g instead of g for the element of A [ G ]. Thus, a Crossed products by actions of finite groups: Some examples. general element of A [ G ] has the form c = � g ∈ G c g u g with c g ∈ A for The Rokhlin property for actions of finite groups. g ∈ G . The multiplication and adjoint are given by: Examples of actions with the Rokhlin property. ( au g )( bu h ) = ( a [ u g bu − 1 Crossed products of AF algebras by actions with the Rokhlin property. g ]) u gh = ( a α g ( b )) u gh Other crossed products by actions with the Rokhlin property. ( au g ) ∗ = u ∗ g a ∗ = ( u − 1 g a ∗ u g ) u − 1 = α − 1 g ( a ∗ ) u g − 1 . g The tracial Rokhlin property for actions of finite groups. for a , b ∈ A and g , h ∈ G , extended linearly. In particular, u ∗ g = u g − 1 . Examples of actions with the tracial Rokhlin property. Crossed products by actions with the tracial Rokhlin property. Exercise: Prove that these definitions make A [ G ] a *-algebra over C . Applications of the tracial Rokhlin property. There is a unique norm which makes this a C*-algebra. (See below.) N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 3 / 29 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 4 / 29

Crossed products by finite groups (continued) Crossed products by finite groups (continued) Let α : G → Aut( A ) be an action of a finite group G on a C*-algebra A . Recall: To keep as elementary as possible, assume that A is unital. We construct ( au g ) ∗ = α − 1 g ( a ∗ ) u g − 1 . ( au g )( bu h ) = a α g ( b ) u gh and a C* norm on the skew group ring A [ G ]. Also, for c = � g ∈ G c g u g , Recall: � ( au g ) ∗ = α − 1 π ( α − 1 g ( a ∗ ) u g − 1 . ( σ ( c ) ξ ) h = h ( c g ))( ξ g − 1 h ) . ( au g )( bu h ) = ( a α g ( b )) u gh and g ∈ G Fix a unital faithful representation π : A → L ( H 0 ) of A on a Hilbert For a ∈ A and g ∈ G , identify a with au 1 and get space H 0 . Set H = l 2 ( G , H 0 ), the set of all ξ = ( ξ g ) g ∈ G in � g ∈ G H 0 , ( σ ( a ) ξ ) h = π ( α h − 1 ( a ))( ξ h ) and ( σ ( u g ) ξ ) h = ξ g − 1 h . with the scalar product � � � ( ξ g ) g ∈ G , ( η g ) g ∈ G = � ξ g , η g � . One can check that σ is a *-homomorphism. We will just check the most important part, which is that σ ( u g ) σ ( b ) = σ ( α g ( b )) σ ( u g ). We have g ∈ G � � � � Then define σ : A [ G ] → L ( H ) as follows. For c = � σ ( α g ( b )) σ ( u g ) ξ h = π α h − 1 ( α g ( b )) ( σ ( u g ) ξ ) h = π ( α h − 1 g ( b ))( ξ g − 1 h ) g ∈ G c g u g , and � π ( α − 1 ( σ ( c ) ξ ) h = h ( c g ))( ξ g − 1 h ) � � � � � � � σ ( u g ) σ ( b ) ξ h = σ ( b ) ξ g − 1 h = π α h − 1 g ( b ) ξ g − 1 h . g ∈ G Exercise: Prove in detail that σ , as defined above, is a *-homomorphism. for all h ∈ G . (Some explanation is on the next slide.) N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 5 / 29 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 6 / 29 Crossed products by finite groups (continued) Crossed products by finite groups (continued) Recall: π : A → L ( H 0 ) is an isometric representation of A , H = � g ∈ G H 0 , We are still considering an action α : G → Aut( A ) of a finite group G on a and σ : A [ G ] → L ( H ) is given, for c = � g ∈ G c g u g ∈ A [ G ] and C*-algebra A . ξ = ( ξ g ) g ∈ G ∈ H , by We started with a faithful representation π : A → L ( H 0 ) of A on a Hilbert space H 0 . Then we constructed a representation σ : A [ G ] → L ( l 2 ( G , H 0 )), � π ( α − 1 ( σ ( c ) ξ ) h = h ( c g ))( ξ g − 1 h ) . given, for c = � g ∈ G c g u g ∈ A [ G ] and ξ = ( ξ g ) g ∈ G ∈ H , by g ∈ G � π ( α − 1 ( σ ( c ) ξ ) h = h ( c g ))( ξ g − 1 h ) . It is easy to check that g ∈ G � � σ ( c ) � ≤ � c g � . We found that A [ G ] is complete in the norm � c � = � σ ( c ) � . By standard g ∈ G theory, the norm � c � = � σ ( c ) � is therefore the only norm in which A [ G ] is Exercise: Prove this. a C*-algebra. In particular, it does not depend on the choice of π . Exercise: Prove that � σ ( c ) � ≥ max g ∈ G � c g � . We return to the notation C ∗ ( G , A , α ) for the crossed product. Hint: Look at σ ( c ) ξ for ξ in just one of the summands of H 0 in H , that is, Things are more complicated if G is discrete but not finite. (In particular, ξ k = 0 for all but one k ∈ G . there may be more than one reasonable norm—since A [ G ] isn’t complete, The norms on the right hand sides are equivalent, so A [ G ] is complete in this is not ruled out.) The situation is even more complicated if G is the norm � c � = � σ ( c ) � . merely locally compact. N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 7 / 29 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 27 July 2014 8 / 29