Regular Ideals, Ideal Intersections, and Quotients

Let \(B \subseteq A\) $B\subseteq A$ be an inclusion of \(C^*\) $C^{\ast }$ -algebras. We study the relationship between the regular ideals of B and regular ideals of A. We show that if \(B \subseteq A\) $B\subseteq A$ is a regular \(C^*\) $C^{\ast }$ -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 \(D \subseteq A\) $D\subseteq A$ is a Cartan inclusion and J is a regular ideal in A, then \(D/(J\cap D)\) $D/(J\cap D)$ is a Cartan subalgebra of A/J. We provide a description of regular ideals in the reduced crossed product of a C \(^*\) ${}^{\ast }$ -algebra by a discrete group.