The ideal structure of measure algebras and asymptotic properties of group representations

TL;DR

This paper classifies weak*-closed maximal left ideals of measure algebras for certain groups, revealing their structure via irreducible representations and asymptotic properties, with applications to representation theory and ideal generation.

Contribution

It provides a classification of maximal left ideals in measure algebras for specific groups and explores their properties, including weak*-closedness and finite generation.

Findings

Classification for connected nilpotent Lie groups

Identification of weak*-closed maximal ideals in Euclidean motion groups

Existence of weak*-closed maximal ideals not generated by projections

Abstract

We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We also prove a version of this result for certain weighted measure algebras. We apply our classification to obtain an analogue of Barnes' Theorem on integrable representations for representations vanishing at infinity. We next study the relationship between weak*-closedness and finite generation, proving that in many cases $M(G)$ has no finitely-generated maximal left ideals. We also show that the measure algebra of the 2D Euclidean rigid motion group has a weak*-closed maximal left ideal that is not generated by a projection, and investigate whether or not it has any weak*-closed left ideals which are not finitely-generated.