arXiv subjects

Sarah A. Reznikoff

Publications and source records attributed to Sarah A. Reznikoff.

4 recordsLinked to original sources

arXiv2026

Regular ideals, Ideal Intersections and Quotients II

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.

math.OA↗

arXiv2023

Regular ideals, ideal intersections, and quotients

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 $A \rtimes_r Γ$.

math.OA↗

arXiv2020

Regular ideals of graph algebras

Let $C^*(E)$ be the graph C$^*$-algebra of a row-finite graph $E$. We give a complete description of the vertex sets of the gauge-invariant regular ideals of $C^*(E)$. It is shown that when $E$ satisfies Condition (L) the regular ideals $C^*(E)$ are a class of gauge-invariant ideals which preserve Condition (L) under quotients. That is, we show that if $E$ satisfies Condition (L) then a regular ideal $J \unlhd C^*(E)$ is necessarily gauge-invariant. Further, if $J \unlhd C^*(E)$ is a regular ideal, it is shown that $C^*(E)/J \simeq C^*(F)$ where $F$ satisfies Condition (L).

math.OA↗

arXiv2019

Intermediate C*-algebras of Cartan Embeddings

Let $A$ be a C$^*$-algebra and let $D$ be a Cartan subalgebra of $A$. We study the following question: if $B$ is a C$^*$-algebra such that $D \subseteq B \subseteq A$, is $D$ a Cartan subalgebra of $B$? We give a positive answer in two cases: the case when there is a faithful conditional expectation from $A$ onto $B$, and the case when $A$ is nuclear and $D$ is a C$^*$-diagonal of $A$. In both cases there is a one-to-one correspondence between the intermediate C$^*$-algebras $B$, and a class of open subgroupoids of the groupoid $G$, where $Σ\rightarrow G$ is the twist associated with the embedding $D \subseteq A$.

math.OA↗