Groupoids and 𝐶*-algebras for categories of paths

Spielberg, Jack

doi:10.1090/S0002-9947-2014-06008-X

Groupoids and $C^*$-algebras for categories of paths
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Trans. Amer. Math. Soc. 366 (2014), 5771-5819 Request permission

Abstract:

In this paper we describe a new method of defining $C^*$-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the $C^*$-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.

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