We prove that every normal coaction of a discrete group on an operator algebra extends to a (unique) normal coaction on the C*-envelope of the operator algebra.

Left cancellative small categories — viewed as generalizations of monoids — provide a natural and vastly general framework for studying operator algebras generated by partial isometries. All left cancellative monoids and higher-rank graphs fit into this setting, but it is much more general. Using our extension theorem combined with new groupoid-theoretic techniques, we compute the C*-envelope for operator algebras arising from several classes of such small categories, answering a question posed by Xin Li.

This is 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.