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arXiv · 2208.09943

Regular ideals, ideal intersections, and quotients

Jonathan H. Brown·Adam H. Fuller·David R. Pitts·Sarah A. Reznikoff
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Abstract

Let $B \subseteq A$ 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 $B \subseteq A$ 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 $D \subseteq A$ is a Cartan inclusion and $J$ is a regular ideal in $A$, then $D/(J\cap D)$ is a Cartan subalgebra of $A/J$. We provide a description of regular ideals in reduced crossed products $A \rtimes_r \Gamma$.

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Jonathan H. Brown, Adam H. Fuller, David R. Pitts, Sarah A. Reznikoff. 2022-08-21. Regular ideals, ideal intersections, and quotients. https://arxiv.org/abs/2208.09943

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