Intermediate C*-algebras of Cartan Embeddings

TL;DR

This paper investigates when intermediate C*-algebras between a C*-algebra and its Cartan subalgebra retain the Cartan property, establishing conditions for this and describing their structure via groupoid subgroupoids.

Contribution

It provides new criteria ensuring intermediate C*-algebras inherit the Cartan property and characterizes them through open subgroupoids of associated groupoids.

Findings

Positive answer when a faithful conditional expectation exists

Cartan property preserved in nuclear C*-algebras with C*-diagonals

Correspondence between intermediate algebras and open subgroupoids

Abstract

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 $D \subseteq B \subseteq A$, 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$^*$-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 $\Sigma \rightarrow G$ is the twist associated with the embedding $D \subseteq A$.