PhD Thesis Defence

Thursday, July 21, 2022 1:00 pm - 1:00 pm EDT (GMT -04:00)

Adam Humeniuk, Department of Pure Mathematics, University of Waterloo

"Dilation methods in semigroup dynamics and noncommutative convexity"

I will give an exposition of my thesis in two parts. Part one concerns C*-envelopes of non-selfadjoint operator algebras associated to semigroup dynamics. We show that the C*-envelope of a Nica-covariant semicrossed product over a lattice ordered abelian semigroup is a full corner of a crossed product, by dilating the semigroup action to an action of the group it generates. This extends results of Davidson, Fuller, and Kakariadis to a large class of abelian semigroups.

Part two concerns noncommutative ("nc" or "matrix") convexity, which is dual to the study of operator systems. Firstly, We show that a multivariable nc function which is convex in each variable satisfies a noncommutative generalization of the classical Jensen's inequality. This yields a connection to free probability. Secondly, we discuss duality for operator systems through the lens of nc convexity. We characterize when the dual of an operator system is again an operator system in terms of geometric conditions on its associated nc convex set. These results are a geometric counterpart to a recent algebraic characterization by C.K. Ng.

Online. Please contact Adam Humeniuk at for more information.