Regular ideals, ideal intersections, and quotients

Let $B\subseteq A$ be an inclusion of C${}^{\ast }$-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${}^{\ast }$-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 $D\mathrm{/}(J\cap D)$ is a Cartan subalgebra of $A\mathrm{/}J$. We provide a description of regular ideals in reduced crossed products $A{⋊}_r\mathrm{\Gamma }$.