Left cancellative small categories — viewed as generalizations of monoids — give rise to reduced and universal Toeplitz C*-algebras, each of which contains a canonical non-selfadjoint operator algebra, and to reduced and universal boundary quotient C*-algebras. The theory of C*-algebras from such categories was initiated by Spielberg; it unifies the theories of semigroup and (higher rank) graph C*-algebras, but also contains many other interesting classes of operator algebras.

I will explain how to construct the reduced Toeplitz and boundary quotient C*-algebras from left cancellative small categories and how to describe these C*-algebras using dynamics. The talk is mostly expository: I want to advertise operator algebras from left cancellative small categories as a basic, natural, and unifying language for studying many selfadjoint and non-selfadjoint operator algebras. However, I will mention a recent contribution to the theory from joint work with Kevin Aguyar Brix and Adam Dor-On.

Bio: Chris Bruce is a Marie Curie Fellow at the University of Glasgow working on operator algebras and interactions with etale groupoids, algebraic dynamics, and algebraic number theory. Chris completed his PhD at the University of Victoria in 2020 supervised by Professor Marcelo Laca, and he was an NSERC Banting Fellow with Queen Mary, University of London for two years before moving to Glasgow.