Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras

TL;DR

This paper explores the structure of intermediate subalgebras in Cartan embeddings within rings and C*-algebras, establishing a correspondence with subgroupoids and characterizing when subalgebras preserve Cartan properties.

Contribution

It introduces a lattice isomorphism between subgroupoids and subalgebras in quasi-Cartan pairs, and characterizes when intermediate subalgebras retain Cartan or diagonal structures.

Findings

Lattice isomorphism between subgroupoids and subalgebras in quasi-Cartan pairs

Characterization of subalgebras preserving Cartan properties in rings and C*-algebras

Complete description of Cartan pairs with all intermediate subalgebras also being Cartan

Abstract

Let $D \subseteq A$ be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist $\Sigma \to G$ whose twisted Steinberg algebra is isomorphic to $A$ in a way that preserves $D$. In this paper, we show there is a lattice isomorphism between wide open subgroupoids of $G$ and subalgebras $C$ such that $D\subseteq C\subseteq A$ and $D \subseteq C$ is a quasi-Cartan pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a diagonal/Cartan/quasi-Cartan pair. In the diagonal case, when the coefficient ring is a field, it is all of them. Beyond that, only pairs that are close to being diagonal have this property. We then apply our techniques to C*-algebraic inclusions and give a complete characterization of which Cartan pairs $D \subseteq A$ have the property that every C*-subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a Cartan pair.