Regular ideals, Ideal Intersections and Quotients II

Abstract

Let $B\subseteq A$ be a regular inclusion of C*-algebras satisfying the ideal intersection property and with a faithful invariant pseudo-expectation. A complete description of the regular ideals of $A$ is given using the invariant regular ideals of $B$ and the pseudo-expectation. Further, necessary and sufficient conditions are given for a quotient by a regular ideal to preserve the faithful unique pseudo-expectation property. Special attention is given throughout to pseudo-Cartan inclusions, i.e. regular inclusions with the faithful unique pseudo-expectation property, equivalently, having a Cartan envelope. We show that the quotient of a pseudo-Cartan inclusion by a regular ideal is again a pseudo-Cartan inclusion, and we describe the Cartan envelope of the quotient.

We have added Sections 5 and 6. Section 5 uses reduced crossed products to construct pseudo-Cartan inclusions which are not Cartan inclusions, and Section 6 gives examples which illustrate our results. 38 pages