Gaute Schwartz: Groupoid models of semigroup C*-algebras

Abstract: for a left-cancellative semigroup, \( S \), one always has an associated reduced C*-algebra \( C_r^*(S) \), but defining a universal counterpart can be challenging. One approach is to find an étale groupoid, \( G(S) \), such that \( C_r^*(S) \) is isomorphic to \( C_r^*(G(S)) \)- the reduced C*-algebra of \( G(S) \). Then \( G(S) \) is called a groupoid model for \( S \) and once we have it, it makes sense to define the universal C*-algebra of \( S \) as the universal C*-algebra of \( G(S) \). In this talk we go through the construction of one such groupoid model, the so called reduced Paterson groupoid. We also explain how a subgroupoid of this one can serve as a groupoid model for the boundary quotient of \( S \). To motivate things we focus on the free monoid on \( n \) generators. This is based on joint work with Sergey Neshveyev.