Intermediate C*-algebras of Cartan embeddings

Abstract

Let A A be a C ∗ ^* -algebra and let D D be a Cartan subalgebra of A A . We study the following question: if B B is a C ∗ ^* -algebra such that D ⊆ B ⊆ A D \subseteq B \subseteq A , is D D a Cartan subalgebra of B B ? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A A onto B B , and the case when A A is nuclear and D D is a C ∗ ^* -diagonal of A A . In both cases there is a one-to-one correspondence between the intermediate C ∗ ^* -algebras B B , and a class of open subgroupoids of the groupoid G G , where Σ → G \Sigma \rightarrow G is the twist associated with the embedding D ⊆ A D \subseteq A .