INTERMEDIATE C∗-ALGEBRAS OF CARTAN EMBEDDINGS
Let A be a C∗-algebra and let D be a Cartan subalgebra of A. We study the following question: if B is a C∗-algebra such that D ⊆ B ⊆ A, is D a Cartan subalgebra of B? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A onto B, and the case when A is nuclear and D is a C∗-diagonal of A. In both cases there is a one-to-one correspondence between the intermediate C∗-algebras B, and a class of open subgroupoids of the groupoid G, where Σ → G is the twist associated with the embedding D ⊆ A.
Brown, Jonathan H.; Exel, Ruy; Fuller, Adam H.; Pitts, David R.; and Reznikoff, Sarah A., "INTERMEDIATE C∗-ALGEBRAS OF CARTAN EMBEDDINGS" (2021). Mathematics Open Access Publications. 31.