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arXiv · 1804.00967

Groupoid Models of $C^*$-algebras and Gelfand Duality

Kyle Austin·Atish Mitra
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Abstract

We construct a large class of morphisms, which we call partial morphisms, of groupoids that induce $*$-morphisms of maximal and minimal groupoid $C^*$-algebras. We show that the association of a groupoid to its maximal (minimal) groupoid $C^*$-algebra and the association of a partial morphism to its induced morphism are functors (both of which extend the Gelfand functor). We show how to geometrically visualize lots of $*$-morphisms between groupoid $C^*$-algebras. As an application, we construct a groupoid models of the entire inductive systems of the Jiang-Su algebra $\mathcal{Z}$ and the Razak-Jacelon algebra $\mathcal{W}$.

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Kyle Austin, Atish Mitra. 2018-04-03. Groupoid Models of $C^*$-algebras and Gelfand Duality. https://arxiv.org/abs/1804.00967

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