Inversion problem in measure and Fourier-Stieltjes algebras

Abstract: Let $G$ be a locally compact Abelian group with its dual $\widehat{G}$ and let $M(G)$ denote the Banach algebra of complex-valued measures on $G$. The classical Wiener-Pitt phenomenon asserts that the spectrum of a measure may be strictly larger than the closure of the range of its Fourier-Stieltjes transform. In particular, if $G$ is non-discrete, there exists µ ∈ $M(G)$ such that |$\widehat{\mu}$(γ)| > c > 0 for every γ ∈ $\widehat{G}$ but µ is not invertible. In the paper [N], N. Nikolski suggested the following problem.

Problem 1. Let µ ∈ $M(G)$ satisfy $\|\mu\|$ ≤ 1 and |$\widehat{\mu}$(γ)| ≥ δ for every γ ∈ $\widehat{G}$. What is the minimal value of $\delta_0$ assuring the invertibility of µ for every δ > $\delta_0$? What can be said about the inverse (in terms of δ)?

In my talk I show that $\delta_0 =1/2$ is the optimal value for the first question (for non-discrete $G$). Also, I will present a partial solution for the quantitative variant of the problem (second question): if all elements of G (except the unit) are of infinite order then we can control the norm of the inverse for every δ > $\frac{-1+\sqrt{33}}{8}$. This improves the original result of Nikolski: δ > $\frac{1}{\sqrt{2}}$. If time permits I will present some generalizations of the aformentioned results for FourierStieltjes algebras built on non-commutative groups. The talk is based on a paper [OW] written in collaboration with Mateusz Wasilewski.

functional analysisgroup theoryoperator algebrasquantum algebra

Audience: researchers in the topic

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