Proof of the Ghahramani-Lau conjecture

TL;DR

This paper proves the Ghahramani-Lau conjecture, demonstrating that the measure algebra of any locally compact or Polish group is strongly Arens irregular, by introducing new classes of measures.

Contribution

It establishes the conjecture for all locally compact and Polish groups and introduces new classes of measures of independent interest.

Findings

Measure algebra of locally compact groups is strongly Arens irregular

The result extends to Polish groups

New classes of measures: approximately invariant and strongly singular

Abstract

The Ghahramani-Lau conjecture is established; in other words, the measure algebra of every locally compact group is strongly Arens irregular. To this end, we introduce and study certain new classes of measures (called approximately invariant, respectively, strongly singular) which are of interest in their own right. Moreover, we show that the same result holds for the measure algebra of any (not necessarily locally compact) Polish group.