Abstract

In this talk, I will describe some results obtained jointly with J. Brown, A. Fuller, and S. Reznikoff.   Let $\mathcal A$ be a $C^{$-algebra and $X\subseteq \mathcal A$ and let $X}\perp=\{a\in \mathcal A: aX=Xa=\{0\}\}$. An ideal $J$ in a $C$-algebra $\mathcal A$ is regular if $(J\perp)^\perp=J$. Hamana has shown that the family of regular ideals in $\mathcal A$ forms a Boolean algebra.   Now let $(\mathcal{A},\mathcal{B})$ be a pair of $C\sp *$-algebras with $\mathcal{B}\subseteq\mathcal{A}$ and assume $\mathcal B$ contains an approximate unit for $\mathcal A$.   When the inclusion $(\mathcal A,\mathcal B)$ has a faithful invariant pseudo-expectation, I will discuss a description of the regular ideals of $\mathcal A$ in terms of the invariant regular ideals of $\mathcal B$ and the pseudo-expectation.   I will also discuss the following results. If $(\mathcal A,\mathcal B)$ is a Cartan inclusion and $J$ is a regular ideal in $\mathcal A$, the quotient $(\mathcal A/J, \mathcal B/(\mathcal B\cap J))$ is again a Cartan inclusion. This result can be extended to the class of pseudo-Cartan inclusions.