Jonathan Brown (University of Dayton) Canonical Cartan Subalgebras of Étale Groupoid $C}^{*$-Algebras and the Construction of Twists Sponsored by the Meyer Fund

Dec. 12, 2017 12:10pm (MATH …

Kempner

Kenneth McLaughlin (CSU) Random matrices, d-bar problems, and approximation theory

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Let $G$ be an étale groupoid. In joint work with Nagy, Reznikoff, Sims and Williams, we study a subgroupoid $G\prime$ of $G$ consisting of isotropy, in an effort to determine when a representation of ${C}^{*}(G)$ is injective. We show that a representation is injective on ${C}^{*}(G)$ if and only if its restriction to ${C}^{*}(G\prime )$ is, and that when $G\prime$ is closed, then ${C}^{*}(G\prime )$ is a Cartan subalgebra of ${C}^{*}(G)$. Now, Renault in 2008 showed that given a Cartan subalgebra $D$ of a $C}^{*$-algebra $A$, one can construct a twist $T$ so that ${C}^{*}(T)$ is isomorphic to $A$. In this talk, I will first describe my joint results with Nagy et al. and then show how the work of Muhly, Renault and Williams can be used to explicitly construct a twist associated to the inclusion of ${C}^{*}(G\prime )$ into ${C}^{*}(G)$ when $G\prime$ is all of the isotropy. I will then point out how this fails more generally.

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Some surprising questions in analysis arise in the interconnections of the topics in the title. We will encounter zeros of the Taylor approximants of exp(z), and other analytic functions. We will consider questions of support of equilibrium charge distributions in the plane. Semi-classical analysis of d-bar problems will provide merriment along the way.