Event Details

A mathematics talk for a general audience, given by a member of the School of Mathematical Sciences, Dr. Yemon Choi.

Title: Does left invertibility imply invertibility for convolution algebras of groups?

Abstract: If V is a finite-dimensional vector space and S, T are linear maps from V to itself satisfying TS=I, then a dimension-counting argument shows that ST=I. This property fails for infinite dimensional vector spaces. Nevertheless, if G is any group and one considers its complex group algebra CG, then any pair of elements a,b in CG satisfying ba=1 must satisfy ab=1. Perhaps surprisingly, all known proofs require ideas and tools from analysis (albeit sometimes hidden away from polite company).

In this talk I will attempt to outline why the result is true, and then discuss various possible generalizations when G is a "continuous group" and CG is replaced by certain "continuous analogues". Most of the talk should be accessible to students who have seen the

definitions of groups, rings, metric spaces, and Hilbert spaces, although we will have to take certain results from postgraduate-level-analysis on trust. Towards the end some familiarity with the Y4 Operator Theory course (or equivalent background) would be helpful.