Synthetic properties of locally compact groups: preservation and transference

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.