Synthetic properties of locally compact groups: preservation and transference

TL;DR

This paper investigates how certain properties related to operator synthesis and sets are preserved or transferred between locally compact groups via group homomorphisms, extending existing theoretical results.

Contribution

It establishes new transference and preservation results for operator sets under group homomorphisms, broadening the understanding of properties in harmonic analysis.

Findings

Preservation of compact operator synthesis sets under specific group homomorphisms

Extension of results to operator Ditkin sets and M-sets

Generalization of existing theorems in harmonic analysis and operator theory

Abstract

Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $\alpha$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure on $H$, then $(\alpha\times\alpha)^{-1}$ preserves sets of compact operator synthesis, and conversely when $\alpha$ is onto. We also prove similar preservation results for operator Ditkin sets and operator M-sets, obtaining preservation results for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.