# Reduced spectral synthesis and compact operator synthesis

## TL;DR

This paper introduces the concept of reduced spectral synthesis, unifying spectral synthesis and uniqueness in locally compact groups, and explores its operator algebraic counterpart, linking it to operator equations and exceptional sets.

## Contribution

It defines reduced spectral synthesis, establishes its properties, and connects it with compact operator synthesis, providing new insights into spectral and operator algebra theory.

## Key findings

- Non-discrete groups with open abelian subgroups have subsets failing reduced spectral synthesis.
- Reduced local spectral synthesis is equivalent to compact operator synthesis for related sets.
- Applications to operator equations with normal commuting coefficients on Schatten classes.

## Abstract

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 with 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 $E^* = \{(s,t) : ts^{-1}\in E\}$ of $G\times G$ 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.