The Second Summer School on Operator Algebras and Noncommutative Geometry 2016 East China Normal University, Shanghai Lecture 3: Crossed Products by Finite Groups; the 11–29 July 2016 Rokhlin Property 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 15 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; the Rokhlin Property 15 July 2016 1 / 39 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 2 / 39 A rough outline of all six lectures Recall: Group actions on C*-algebras The beginning: The C*-algebra of a group. Actions of finite groups on C*-algebras and examples. Definition Crossed products by actions of finite groups: elementary theory. Let G be a group and let A be a C*-algebra. An action of G on A is a More examples of actions. homomorphism g �→ α g from G to Aut( A ). Crossed products by actions of finite groups: Some examples. The Rokhlin property for actions of finite groups. That is, for each g ∈ G , we have an automorphism α g : A → A , and Examples of actions with the Rokhlin property. α 1 = id A and α g ◦ α h = α gh for g , h ∈ G . Crossed products of AF algebras by actions with the Rokhlin property. When G is a topological group, we require that the action be continuous: Other crossed products by actions with the Rokhlin property. ( g , a ) �→ α g ( a ) is jointly continuous from G × A to A . The tracial Rokhlin property for actions of finite groups. Examples of actions with the tracial Rokhlin property. Crossed products by actions with the tracial Rokhlin property. Applications of the tracial Rokhlin property. N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 3 / 39 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 4 / 39
Recall: The action of SL 2 ( Z ) on the torus Reminder: The rotation algebras Let θ ∈ R . Recall the irrational rotation algebra A θ , the universal Recall: Every action of a group G on a compact space X gives an action C*-algebra generated by two unitaries u and v satisfying the commutation relation vu = e 2 π i θ uv . Some standard facts, presented without proof: of G on C ( X ). If θ �∈ Q , then A θ is simple. In particular, any two unitaries u and v in The group SL 2 ( Z ) acts on R 2 via the usual matrix multiplication. This any C*-algebra satisfying vu = e 2 π i θ uv generate a copy of A θ . action preserves Z 2 , and so is well defined on R 2 / Z 2 ∼ = S 1 × S 1 . If θ = m n in lowest terms, with n > 0, then A θ is isomorphic to the SL 2 ( Z ) has finite cyclic subgroups of orders 2, 3, 4, and 6, generated by section algebra of a locally trivial continuous field over S 1 × S 1 with � − 1 � � − 1 � � 0 � � 0 � 0 − 1 − 1 − 1 fiber M n . , , , and . 0 − 1 1 0 1 0 1 1 = C ( S 1 × S 1 ). In particular, if θ = 0, or if θ ∈ Z , then A θ ∼ The algebra A θ is often considered to be a noncommutative analog of the Restriction gives actions of these on S 1 × S 1 . torus S 1 × S 1 (more accurately, a noncommutative analog of C ( S 1 × S 1 )). N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 5 / 39 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 6 / 39 The action of SL 2 ( Z ) on the rotation algebra The action of SL 2 ( Z ) on the rotation algebra (continued) Recall: A θ is the universal C*-algebra generated by two unitaries u and v Recall: A θ is the universal C*-algebra generated by two unitaries u and v satisfying the commutation relation vu = e 2 π i θ uv . satisfying the commutation relation vu = e 2 π i θ uv . The group SL 2 ( Z ) acts on A θ by sending the matrix Recall that SL 2 ( Z ) has finite cyclic subgroups of orders 2, 3, 4, and 6, generated by � n 1 , 1 � n 1 , 2 � − 1 � � − 1 � � 0 � � 0 � n = 0 − 1 − 1 − 1 n 2 , 1 n 2 , 2 , , , and . 0 − 1 1 0 1 0 1 1 to the automorphism determined by Restriction gives actions of these groups on the irrational rotation algebras. α n ( u ) = exp( π in 1 , 1 n 2 , 1 θ ) u n 1 , 1 v n 2 , 1 In terms of generators of A θ , and omitting the scalar factors (which are not necessary when one restricts to these subgroups), the action of Z 2 is and generated by α n ( v ) = exp( π in 1 , 2 n 2 , 2 θ ) u n 1 , 2 v n 2 , 2 . u �→ u ∗ v �→ v ∗ , and Exercise: Check that α n is an automorphism, and that n �→ α n is a group and the action of Z 4 is generated by homomorphism. v �→ u ∗ . u �→ v and This action is the analog of the action of SL 2 ( Z ) on S 1 × S 1 = R 2 / Z 2 . It Exercise: Find the analogous formulas for Z 3 and Z 6 , and check that they give actions of these groups. reduces to that action when θ = 0. N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 7 / 39 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 8 / 39
Another example: The tensor flip The free flip Let A be a C*-algebra, and let A ⋆ A be the free product of two copies Assume (for convenience) that A is nuclear and unital. Then there is an of A . (Use A ⋆ C A to get a unital C*-algebra.) Then there is an action of Z 2 on A ⊗ A generated by the “tensor flip” a ⊗ b �→ b ⊗ a . automorphism α ∈ Aut( A ⋆ A ) which exchanges the two free factors. For Similarly, the symmetric group S n acts on A ⊗ n . a ∈ A , it sends the copy of a in the first free factor to the copy of the same element in the second free factor, and similarly the copy of a in the The tensor flip on the 2 ∞ UHF algebra A = � ∞ n =1 M 2 turns out to be second free factor to the copy of the same element in the first free factor. essentially the product type action generated by This automorphism might be called the “free flip”. It generates a actions of Z 2 on A ⋆ A and A ⋆ C A . 1 0 0 0 ∞ ∞ There are many generalizations. One can take the amalgamated free 0 1 0 0 � � Ad on M 4 . product A ⋆ B A over a subalgebra B ⊂ A (using the same inclusion in both 0 0 1 0 n =1 n =1 copies of A ), or the reduced free product A ⋆ r A (using the same state on 0 0 0 − 1 both copies of A ). There is a permutation action of S n on the free product Exercise: Prove this. (Hint: Look at the tensor flip on M 2 ⊗ M 2 .) of n copies of A . And one can make any combination of these generalizations. Another interesting example is gotten by taking A to be the Jiang-Su algebra Z . It is simple, separable, unital, and nuclear. It has no nontrivial See the appendix for some actions on Cuntz algebras, along with a projections, its Elliott invariant is the same as for C , and Z ⊗ Z ∼ = Z . reminder of the definition of Cuntz algebras. N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 9 / 39 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 10 / 39 Recall: Construction of the crossed product by a finite Examples of crossed products by finite groups group Let G be a finite group, and let ι : G → Aut( C ) be the trivial action, defined by ι g ( a ) = a for all g ∈ G and a ∈ C . Then C ∗ ( G , C , ι ) = C ∗ ( G ), Let α : G → Aut( A ) be an action of a finite group G on a unital the group C*-algebra of G . (So far, G could be any locally compact C*-algebra A . As a vector space, C ∗ ( G , A , α ) is the group ring A [ G ], group.) consisting of all formal linear combinations of elements in G with Since we are assuming that G is finite, C ∗ ( G ) is a finite dimensional coefficients in A : C*-algebra, with dim( C ∗ ( G )) = card( G ). If G is abelian, so is C ∗ ( G ), so � � � C ∗ ( G ) ∼ = C card( G ) . If G is a general finite group, C ∗ ( G ) turns out to be A [ G ] = c g · u g : c g ∈ A for g ∈ G . the direct sum of matrix algebras, one summand M k for each unitary g ∈ G equivalence class of k -dimensional irreducible representations of G . The multiplication and adjoint are given by: Now let A be any C*-algebra, and let ι A : G → Aut( A ) be the trivial action. It is not hard to see that C ∗ ( G , A , ι A ) ∼ = C ∗ ( G ) ⊗ A . The elements ( a · u g )( b · u h ) = ( a [ u g bu − 1 g ]) · u g u h = ( a α g ( b )) · u gh of A “factor out”, since A [ G ] is just the ordinary group ring: ( a · u g ) ∗ = α − 1 g ( a ∗ ) · u g − 1 ( a · u g )( b · u h ) = ( a ι g ( b )) · u gh = ( ab ) · u gh . for a , b ∈ A and g , h ∈ G , extended linearly. We saw that there is a unique Exercise: prove this. (Since C ∗ ( G ) is finite dimensional, C ∗ ( G ) ⊗ A is just norm which makes this a C*-algebra. the algebraic tensor product.) N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 11 / 39 N. C. Phillips (U of Oregon) Crossed Products; the Rokhlin Property 15 July 2016 12 / 39
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