Groupoids and $C^*$-algebras for left cancellative small categories

Jack Spielberg

Groupoids and $C^*$-algebras for left cancellative small categories

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arxiv 1712.07720 v2 pith:FXWZUBYF submitted 2017-12-20 math.OA

Jack Spielberg This is my paper · ORCID

classification math.OA

keywords algebrascategoriescancellativegeneralleftsituationsmallalgebra

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abstract

Categories of paths are a generalization of several kinds of oriented discrete data that have been used to construct $C^*$-algebras. The techniques introduced to study these constructions apply almost verbatim to the more general situation of left cancellative small categories. We develop this theory and derive the structure of the $C^*$-algebras in the most general situation. We analyze the regular representation, and the Wiener-Hopf algebra in the case of a subcategory of a groupoid.

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Subshift semigroups
math.OA 2019-08 accept novelty 7.0 of 10

For every subshift, the Matsumoto and Carlsen-Matsumoto C*-algebras are realized as groupoid C*-algebras from the inverse hull of the language semigroup, and this universal groupoid is amenable.

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