Ordered invariant ideals of Fourier-Stieltjes algebras

TL;DR

This paper explores properties of invariant ideals in Fourier-Stieltjes algebras, introduces the concepts of ordered and weakly ordered ideals, and discusses implications for related lemmas and open questions.

Contribution

It defines and initiates the study of ordered and weakly ordered invariant ideals in Fourier-Stieltjes algebras, addressing a gap in previous proofs.

Findings

Introduction of ordered and weakly ordered ideal properties

Analysis of the impact on existing lemmas and theories

Identification of open questions in the structure of Fourier-Stieltjes algebras

Abstract

In a recent paper on exotic crossed products, we included a lemma concerning ideals of the Fourier-Stieltjes algebra. Buss, Echterhoff, and Willett have pointed out to us that our proof of this lemma contains an error. In fact, it remains an open question whether the lemma is true as stated. In this note we indicate how to contain the resulting damage. Our investigation of the above question leads us to define two properties \emph{ordered} and \emph{weakly ordered} for invariant ideals of Fourier-Stieltjes algebras, and we initiate a study of these properties.