Descriptive set-theoretic aspects of closed sets of uniqueness in the non-abelian setting

Artikel i vetenskaplig tidskrift, 2022

We study the family of closed sets of (extended) uniqueness of a locally compact group G which is not necessarily abelian. We prove some preservation properties concerning this family of sets as well as their operator-theoretic counterpart, locate their descriptive complexity and establish sufficient conditions for the non-existence of a Borel basis. 1. Introduction. The study of sets of uniqueness has a long and illus-trious history witnessing fruitful interdisciplinary efforts and the develop-ment of techniques whose range of applicability goes beyond the realm of harmonic analysis. Unequivocally a rich topic, it constitutes a rather fertile soil for unexpected counter-examples and remarkable theorems. Classically, sets of uniqueness (and sets of extended uniqueness, among other thin sets) were studied on T and despite early accomplishments, the task of charac-terizing these sets remained an open problem. For a comprehensive source on sets of uniqueness of the circle, which is accompanied by several digres-sions on these difficulties, the reader is referred to [14]. In the 1950s, the subject regained additional vigour due to a functional-analytic reformula-tion of its language and in [5], the notion of (closed) set of uniqueness is extended to general locally compact groups relying on the framework of Fourier algebras, introduced in [9]. In the 1980s and 1990s, the incorpora-tion of descriptive set-theoretic methods in the study of sets of uniqueness (mostly in the abelian case) provided a deeper insight into some classical questions through the lens of a rather powerful language. In this paper we study some descriptive set-theoretic aspects-such as the descriptive com-plexity and (non-)existence of Borel basis-of closed sets of uniqueness in 2020 Mathematics Subject Classification: Primary 43A46; Secondary 03E15.