# Spectral theory of Fourier-Stieltjes algebras

## TL;DR

This paper explores the spectral properties of Fourier-Stieltjes algebras on discrete groups, revealing phenomena like the Wiener-Pitt effect and developing tools to analyze their spectra and boundary behaviors.

## Contribution

It extends spectral analysis of Fourier-Stieltjes algebras to non-abelian groups, introducing generalized characters and criteria for boundary elements.

## Key findings

- Wiener-Pitt phenomenon occurs in large classes of discrete groups.
- Distinct notions of support arise due to non-commutativity.
- Generalized characters help identify elements of the Shilov boundary.

## Abstract

In this paper we start studying spectral properties of the Fourier-Stieltjes algebras, largely following Zafran's work on the algebra of measures on a locally compact group. We show that for a large class of discrete groups the Wiener-Pitt phenomenon occurs, i.e. the spectrum of an element of the Fourier-Stieltjes algebra is not captured by its range. We also investigate the notions of absolute continuity and mutual singularity in this setting; non-commutativity forces upon us two distinct versions of support of an element, indicating a crucial difference between this setup and the realm of Abelian groups. In spite of these difficulties, we also show that one can introduce and use generalised characters to prove a criterion on belonging of a multiplicative-linear functional to the Shilov boundary of the Fourier-Stieltjes algebra.