Regular Ideals, Ideal Intersections, and Quotients

Let B⊆ 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⊆ 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⊆ A is a Cartan inclusion and J is a regular ideal in A, then D/ (J∩ D) is a Cartan subalgebra of A/J. We provide a description of regular ideals in the reduced crossed product of a C ^∗ -algebra by a discrete group.