Decomposing the C -algebras of groupoid extensions

Transcription

1 Decomposing the C -algebras of groupoid extensions Jonathan Brown, joint with Astrid an Huef Kansas State University Great Plains Operator Theory Symposium May 2013 J. Brown (KSU) Groupoid extensions GPOTS 1 / 6

2 Twisted Group C -algebras Let H be a topological group e.g. H = Z 2 J. Brown (KSU) Groupoid extensions GPOTS 2 / 6

3 Twisted Group C -algebras Let H be a topological group e.g. H = Z 2 with Haar measure λ. λ counting measure J. Brown (KSU) Groupoid extensions GPOTS 2 / 6

4 Twisted Group C -algebras Let H be a topological group e.g. H = Z 2 with Haar measure λ. λ counting measure Let ω : H H T be a 2-cocycle: ω(x, y)ω(xy, z) = ω(x, yz)ω(y, z). J. Brown (KSU) Groupoid extensions GPOTS 2 / 6

5 Twisted Group C -algebras Let H be a topological group e.g. H = Z 2 with Haar measure λ. λ counting measure Let ω : H H T be a 2-cocycle: θ [0, 1] ω(x, y)ω(xy, z) = ω(x, yz)ω(y, z). ω((m, n), (p, q)) = e 2πinpθ J. Brown (KSU) Groupoid extensions GPOTS 2 / 6

6 Twisted Group C -algebras Let H be a topological group e.g. H = Z 2 with Haar measure λ. λ counting measure Let ω : H H T be a 2-cocycle: θ [0, 1] ω(x, y)ω(xy, z) = ω(x, yz)ω(y, z). ω((m, n), (p, q)) = e 2πinpθ Consider C c (H) = {f : H C : f continuous, compact support}. f g(x) = f (y)g(y 1 x)ω(y, y 1 x)dλ(y) and f (x) = f (x 1 ) ω(x, x 1 ) H J. Brown (KSU) Groupoid extensions GPOTS 2 / 6

7 Twisted Group C -algebras Let H be a topological group e.g. H = Z 2 with Haar measure λ. λ counting measure Let ω : H H T be a 2-cocycle: θ [0, 1] ω(x, y)ω(xy, z) = ω(x, yz)ω(y, z). ω((m, n), (p, q)) = e 2πinpθ Consider C c (H) = {f : H C : f continuous, compact support}. f g(x) = f (y)g(y 1 x)ω(y, y 1 x)dλ(y) and f (x) = f (x 1 ) ω(x, x 1 ) H Complete C c (H) in a universal norm to obtain C (H, ω) (twisted). If ω 1 we denote C (H, ω) by C (H) (nontwisted). The nontwisted case is more developed than the twisted case. C (Z 2, ω) is the rotation algebra A θ. J. Brown (KSU) Groupoid extensions GPOTS 2 / 6

8 Groupoids A groupoid is a small category where every morphism is an isomorphism. x. xy y J. Brown (KSU) Groupoid extensions GPOTS 3 / 6

9 Groupoids A groupoid is a small category where every morphism is an isomorphism. x. xy Let G be a groupoid. G (0) is the set of identity morphisms. G is principal if the only morphisms which map an object to itself are the identity morphisms. y J. Brown (KSU) Groupoid extensions GPOTS 3 / 6

10 Groupoids A groupoid is a small category where every morphism is an isomorphism. x. xy Let G be a groupoid. G (0) is the set of identity morphisms. G is principal if the only morphisms which map an object to itself are the identity morphisms. Example A group H with identity e is a groupoid with one object e and morphisms h H which map e to e with composition given by group multiplication. y J. Brown (KSU) Groupoid extensions GPOTS 3 / 6

11 Groupoids A groupoid is a small category where every morphism is an isomorphism. x. xy Let G be a groupoid. G (0) is the set of identity morphisms. G is principal if the only morphisms which map an object to itself are the identity morphisms. Example A group H with identity e is a groupoid with one object e and morphisms h H which map e to e with composition given by group multiplication. Given a set of nice measures {λ u } u G (0) and a continuous 2-cocycle ω on a groupoid G we can define C (G, ω) as in the group case. J. Brown (KSU) Groupoid extensions GPOTS 3 / 6 y

12 A case for C (G, ω). An abelian -subalgebra B of a C -algebra A is Cartan if it is maximal abelian and satisfies some technical conditions (Renault 1980, Definition 4.13). If G is principal and étale and ω is a continuous 2-cocycle, then C 0 (G (0) ) is a Cartan subalgebra of C (G, ω). J. Brown (KSU) Groupoid extensions GPOTS 4 / 6

13 A case for C (G, ω). An abelian -subalgebra B of a C -algebra A is Cartan if it is maximal abelian and satisfies some technical conditions (Renault 1980, Definition 4.13). If G is principal and étale and ω is a continuous 2-cocycle, then C 0 (G (0) ) is a Cartan subalgebra of C (G, ω). Theorem (Renault 1980, Theorem II.4.15) If B A is Cartan, then there exists a principal groupoid G and a cocycle ω and a surjective -homomorphism φ : C (G, ω) A such that φ C0 (G (0) ) : C 0(G (0) ) B is an isomorphism. J. Brown (KSU) Groupoid extensions GPOTS 4 / 6

14 Decomposition Let ω be a continuous 2-cocycle of a groupoid G. Note that ω n : (x, y) ω(x, y) n for n Z is also a continuous 2-cocycle. Define G ω := {(x, t) G T} with multiplication given by (x, t 1 )(y, t 2 ) = (xy, ω(x, y)t 1 t 2 ). J. Brown (KSU) Groupoid extensions GPOTS 5 / 6

15 Decomposition Let ω be a continuous 2-cocycle of a groupoid G. Note that ω n : (x, y) ω(x, y) n for n Z is also a continuous 2-cocycle. Define G ω := {(x, t) G T} with multiplication given by (x, t 1 )(y, t 2 ) = (xy, ω(x, y)t 1 t 2 ). Theorem (B., an Huef) C (G ω ) = n Z C (G, ω n ). J. Brown (KSU) Groupoid extensions GPOTS 5 / 6

16 Decomposition Let ω be a continuous 2-cocycle of a groupoid G. Note that ω n : (x, y) ω(x, y) n for n Z is also a continuous 2-cocycle. Define G ω := {(x, t) G T} with multiplication given by (x, t 1 )(y, t 2 ) = (xy, ω(x, y)t 1 t 2 ). Theorem (B., an Huef) C (G ω ) = n Z C (G, ω n ). Many properties of groupoids are preserved under G G ω. Many properties of C -algebras pass to direct summands. J. Brown (KSU) Groupoid extensions GPOTS 5 / 6

17 Decomposition Let ω be a continuous 2-cocycle of a groupoid G. Note that ω n : (x, y) ω(x, y) n for n Z is also a continuous 2-cocycle. Define G ω := {(x, t) G T} with multiplication given by (x, t 1 )(y, t 2 ) = (xy, ω(x, y)t 1 t 2 ). Theorem (B., an Huef) C (G ω ) = n Z C (G, ω n ). Many properties of groupoids are preserved under G G ω. Many properties of C -algebras pass to direct summands. So our theorem provides a method for proving results about twisted groupoid C -algebras from known results about nontwisted groupoid C -algebras. J. Brown (KSU) Groupoid extensions GPOTS 5 / 6

18 Consequences Suppose G is a second countable locally compact Hausdorff principal groupoid with Haar system and ω is a continuous 2-cocycle. If C (G) is continuous trace bounded trace Fell liminal post-liminal Some implications already known Muhly Williams 1992 Clark, an Huef then C (G, ω) is too. J. Brown (KSU) Groupoid extensions GPOTS 6 / 6