Reduced spectral synthesis and compact operator synthesis
Journal article, 2020

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory, studied previously. We show that a closed subset E of a second countable locally compact group G satisfies reduced local spectral synthesis if and only if the subset E* = {(s, t) : ts(-1) is an element of E} of G x G satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten p-classes. (C) 2020 Elsevier Inc. All rights reserved.

Reduced group C*-algebra

Operator synthesis

Spectral synthesis



V. S. Shulman

Vologda State Technical University

I. G. Todorov

Queen's University Belfast

Nankai University

Lyudmyla Turowska

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 367 107109

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2/5/2021 2