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

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.

The Inverse Hull of 0-Left Cancellative Semigroups

“…Over the past several years, a number of mathematicians have associated C * -algebras to left-cancellative semigroups, or more generally categories [2], [48], [49]. For example, graph C * -algebras (which for finite graphs are basically the same thing as Cuntz-Krieger algebras) are the C * -algebras associated to the path category (or free category) of the graph.…”

Skew products of finitely aligned left cancellative small categories and Cuntz-Krieger algebras

“…LCSC. We refer to [BKQS18,Spi18] for left cancellative small categories and the C * -algebras that may be associated to these. For the special case of "categories of paths", which includes higher-rank graphs, see [Spi14].…”

“…In a recent work [Spi18], generalizing the previous work of many authors (including himself) dealing with directed graphs, higher ranks graphs, categories of paths, and left cancellative monoids, Spielberg has shown how to construct certain groupoids from C and used these to associate a Toeplitz algebra T (C) and a Cuntz-Krieger algebra O(C) to C. For an alternative way of constructing these groupoids, see [OP]. When C is finitely aligned, T (C) and O(C) may be described by generators and relations in a more tractable way than in the general case (see [Spi18,Theorems 9.7 and 10.15]). We will therefore concentrate our attention on the finitely aligned case in the present paper and use these descriptions of T (C) and O(C) as their definitions (cf.…”

The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations

“…The germ of the work described in this paper arose from a discussion on higher rank graphs amongst Nadia Larsen, Mark V. Lawson, Aidan Sims and Alina Vdovina at the end of the 2017 ICMS Workshop Operator algebras: order, disorder and symmetry. This led to ongoing discussions between the two authors centred on the papers [36,19,32,9,38,39]. It quickly became clear that there was a need to find a common language and the present paper was the result.…”