Universidad de Cádiz, España
A decade ago, Spielberg described a new method of defining C*-algebras from oriented combinatorial data, generalizing the construction of algebras from directed graphs, higher-rank graphs, and (quasi-)ordered groups. To this end, he introduced left cancellative small categories, and endowed any such category with a C*-algebra encoding categorical information; he showed that this algebra is the groupoid algebra of a Deaconu-Renault étale groupoid. In this talk, we will try to explain why such algebras are relevant. Also, we will show that they are also Exel's groupoid C*-algebras associated to a suitable inverse semigroup SΛ. Subsequently, we study groupoid actions on left cancellative small categories and their associated Zappa-Szép products using the same strategy. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Szép products. Then, we can characterize properties of them, like being Hausdorff, effective and minimal, and thus simplicity for these algebras. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
The contents of this talk are a joint work with Eduard Ortega (NTNU Trondheim, Norway).