Toronto Set Theory Seminar
Friday, January 13 (next week) from 1:30-3pm
Fields, Room 210
Speaker: Jan Pachl (Fields)
Title: Measurable centres in convolution semigroups
Every topological group G naturally embeds in larger spaces, algebraically and topologically. Two such convolution semigroups of particular interest in abstract harmonic analysis are the norm dual of the space of bounded right uniformly continuous functions on G, and the uniform compactification of G with its right uniformity. Our understanding of the structure of these spaces has been advanced by tractable characterizations of their topological centres, now available for “almost all” topological groups. In the seminar I will discuss a measurable analogue of the topological centre, for various notions of measurability. This notion was investigated by Glasner (2009) for the compactification of a discrete group, using Borel measurability.
The main result is that in convolution semigroups over locally compact groups the Borel-measurable centre coincides with the topological centre [arXiv:1107.3799]. It is an open question whether the same holds for all topological groups. One version of the similar statement in which universal measurability replaces Borel measurability is independent of ZFC.