Erik Séguin, University of Waterloo
Ulam Stability for Quantum Groups and Noncommutative Dynamics
In this talk, we discuss various "noncommutative" extensions of classical commutative phenomena. We focus on two particular cases of this: representation stability for locally compact quantum groups (extending representation stability for classical locally compact groups) and minimality for C*-dynamical systems (extending minimality for topological flows). In the first part of the talk, we discuss the question of Ulam stability for approximate representations of locally compact quantum groups. We show that compact quantum groups and amenable discrete quantum groups are representation stable. In the process, we establish a partial stability result for general amenable locally compact quantum groups, showing that if one imposes additional constraints on the approximate representation in question then the assumption of compactness or discreteness can be dispensed with. In the second part of the talk, we discuss the notion of minimality for C*-dynamical systems. We show that the accepted definition is somewhat unsatisfactory as an extension of its commutative counterpart from both a topological dynamical and an operator algebraic perspective and present an alternative notion of an extension which rectifies these issues. We demonstrate that this alternate notion also encodes operator algebraic structure which does not appear to have a direct dynamical connection, further motivating its study. We outline various characterizations and extension properties for this notion and discuss its consequences.
MC 2009