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Sarah Reznikoff

Publications and source records attributed to Sarah Reznikoff.

6 recordsLinked to original sources
arXiv2020

Cartan subalgebras for non-principal twisted groupoid $C^*$-algebras

Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, Σ)$ for any twist $Σ$ over $G$. However, there are many groupoids which are not topologically principal, yet their (twisted) $C^*$-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle $c$ on $G$ and a subgroupoid $S \subseteq G$ under which $C^*_r(S, c)$ is Cartan in $C^*_r(G, c)$. When $G$ is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

math.OA↗
arXiv2019

Graded C*-algebras and twisted groupoid C*-algebras

Let $C^*$-algebra that is acted upon by a compact abelian group. We show that if the fixed-point algebra of the action contains a Cartan subalgebra $D$ satisfying an appropriate regularity condition, then $A$ is the reduced $C^*$-algebra of a groupoid twist. We further show that the embedding $D \hookrightarrow A$ is uniquely determined by the twist. These results generalize Renault's results on Cartan subalgebras of $C^*$-algebras.

math.OA↗
arXiv2019

Pullbacks of graph C*-algebras from admissible pushouts of graphs

We define an admissible decomposition of a graph $E$ into subgraphs $F_1$ and $F_2$, and consider the intersection graph $F_1\cap F_2$ as a subgraph of both $F_1$ and $F_2$. We prove that, if the graph $E$ is row finite and its decomposition into the subgraphs $F_1$ and $F_2$ is admissible, then the graph C*-algebra $C^*(E)$ of $E$ is the pullback C*-algebra of the canonical surjections from $C^*(F_1)$ and $C^*(F_2)$ onto $C^*(F_1\cap F_2)$.

math.OA↗
arXiv2016

Cartan subalgebras in C*-algebras of Hausdorff etale groupoids

The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff étale groupoid $G$ embeds as a $C^*$-subalgebra $M$ of the reduced $C^*$-algebra of $G$. We prove that the set of pure states of $M$ with unique extension is dense, and deduce that any representation of the reduced $C^*$-algebra of $G$ that is injective on $M$ is faithful. We prove that there is a conditional expectation from the reduced $C^*$-algebra of $G$ onto $M$ if and only if the interior of the isotropy in $G$ is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, $M$ is a Cartan subalgebra. We prove that for a large class of groupoids $G$ with abelian isotropy---including all Deaconu--Renault groupoids associated to discrete abelian groups---$M$ is a maximal abelian subalgebra. In the specific case of $k$-graph groupoids, we deduce that $M$ is always maximal abelian, but show by example that it is not always Cartan.

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arXiv2013

Pseudo-Diagonals and Uniqueness Theorems

We examine a certain type of abelian C*-subalgebras that allow one to give a unified treatment of two uniqueness theorems: for graph C*-algebras and for certain reduced crossed products.

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arXiv2013

A generalized Cuntz-Krieger uniqueness theorem for higher rank graphs

We present a uniqueness theorem for k-graph C*-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C*-algebra, it is sufficient that the representation be injective on a distinguished abelian C*-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C*-algebra, each of which is the unique extension of a state on the distinguished abelian C*-subalgebra.

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