Intermediate C*-algebras of Cartan Embeddings

Abstract

Let $A$ be a C${}^{\ast }$-algebra and let $D$ be a Cartan subalgebra of $A$. We study the following question: if $B$ is a C${}^{\ast }$-algebra such that $D\subseteq B\subseteq A$, 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${}^{\ast }$-diagonal of $A$. In both cases there is a one-to-one correspondence between the intermediate C${}^{\ast }$-algebras $B$, and a class of open subgroupoids of the groupoid $G$, where $\Sigma \to G$ is the twist associated with the embedding $D\subseteq A$.

14 pages