Regular ideals, ideal intersections, and quotients

TL;DR

This paper investigates the structure of regular ideals in C*-algebra inclusions, establishing isomorphisms and properties preserved under quotients, with applications to Cartan subalgebras and crossed products.

Contribution

It provides new insights into the relationship between regular ideals of subalgebras and larger algebras, especially under conditions like faithful invariant expectations.

Findings

Isomorphism between regular ideals of A and invariant regular ideals of B under certain conditions.

Regular ideals in reduced crossed products are characterized.

Cartan subalgebras remain Cartan after quotienting by regular ideals.

Abstract

Let $B \subseteq A$ be an inclusion of C$^*$-algebras. We study the relationship between the regular ideals of $B$ and regular ideals of $A$. We show that if $B \subseteq A$ is a regular C$^*$-inclusion and there is a faithful invariant conditional expectation from $A$ onto $B$, then there is an isomorphism between the lattice of regular ideals of $A$ and invariant regular ideals of $B$. We study properties of inclusions preserved under quotients by regular ideals. This includes showing that if $D \subseteq A$ is a Cartan inclusion and $J$ is a regular ideal in $A$, then $D/(J\cap D)$ is a Cartan subalgebra of $A/J$. We provide a description of regular ideals in reduced crossed products $A \rtimes_r \Gamma$.