Continuity of the Haar dual measure

TL;DR

This paper proves that the dual Haar measure system remains continuous in locally compact abelian group bundles, extending the understanding of measure continuity in the context of group C*-algebras.

Contribution

It demonstrates the lower semi-continuity of Plancherel weights and establishes the continuity of the dual Haar system for locally compact abelian group bundles.

Findings

Lower semi-continuity of Plancherel weights in group bundles

Continuity of the dual Haar system in abelian group bundles

Extension of measure continuity properties in non-trivial group bundle contexts

Abstract

Given a locally compact group bundle, we show that the system of the Plancherel weights of their C*-algebras is lower semi-continuous. As a corollary, we obtain that the dual Haar sytem of a continuous Haar system of a locally compact abelian group bundle is also continuous.