Synthetic properties of locally compact groups: preservation and transference

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 G to absolutely continuous with respect on H, then $$(\alpha \times \alpha )^{-1}$$ preserves sets compact operator synthesis, and conversely onto. also similar preservation results for Ditkin M-sets, obtaining M-sets as corollaries. Some these extend or complement existing Ludwig, Shulman, Todorov Turowska.