A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups

Abstract

Let a discrete group G act on a unital simple C * -algebra A by outer automorphisms. We establish a Galois correspondence H → A ⋊ α,r H between subgroups of G and C *algebras B satisfying A ⊆ B ⊆ A ⋊ α,r G, where A ⋊ α,r G denotes the reduced crossed product. For a twisted dynamical system (A, G, α, σ), we also prove the corresponding result for the reduced twisted crossed product A ⋊ σ α,r G.

Intermediate C*-algebras of Cartan embeddings

“…In Theorem 5.7 we show that there is a one-to-one correspondence between the open subsets of Γ × X and the norm-closed C(X)-bimodules in C(X) Γ. A similar spectral theorem is proved for actions of groups satisfying the approximation property on simple C * -algebras in [10]. This result is also a direct analogue of the Spectral Theorem for Bimodules for Cartan embeddings in von Neumann algebras; see [8,15,21].…”

“…In the Galois correspondence results in [10,12,23] there is always a conditional expectation onto the intermediate algebras A ⊆ B ⊆ A r Γ. In the following theorem we classify the intermediate subalgebras B in Cartan embeddings for which there is a conditional expectation onto B.…”

“…In [10], Cameron and Smith prove a spectral theorem for A-bimodules in A r Γ, where Γ is a discrete group satisfying the approximation property, and A is a simple C * -algebra. There, the bimodules are determined by subsets of Γ.…”

“…In contrast with the work of [10,12,23] there is not necessarily a conditional expectation onto the intermediate subalgebra.…”

“…Similar Galois correspondence results have been proved for reduced crossedproducts of C * -algebras. If A is a C * -algebra and G is a discrete group acting on A by automorphisms, then Choda [12] gives a one-to-one correspondence between subgroups of G and a certain class of intermediate C * -algebras A ⊆ B ⊆ A r G. An intermediate algebra B is in the desired class if, among other things, there is a faithful conditional expectation A r G → B. Cameron and Smith [10], generalising an earlier result of Izumi [23] for finite groups, showed if A is a simple C * -algebra and G is a discrete group acting on A by outer automorphisms, then the map H → A H is a one-to-one correspondence between subgroups of H ⊆ G and C * -algebras B satisfying A ⊆ B ⊆ A G. In this case there is always a faithful conditional expectation from A r G onto an intermediate C * -algebra.…”