On synthetic and transference properties of group homomorphisms

We study Borel homomorphisms θ : G → H for arbitrary locally compact second countable groups G and H for which the measureθ ∗ ( μ ) ( α ) = μ ( θ − 1( α ) )forα ⊆ HaBorel setis absolutely continuous with respect to ν , where μ (respectively, ν ) is a Haar measure for G , (respectively, H ). We define a natural mapping G from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in B ( L 2 ( H ) ) into the class of masa bimodules in B ( L 2 ( G ) ) and we use it to prove that if k ⊆ G × G is a set of operator synthesis, then( θ × θ ) − 1( k )is also a set of operator synthesis and if E ⊆ H is a set of local synthesis for the Fourier algebra A ( H ) , thenθ − 1( E ) ⊆ G is a set of local synthesis for A ( G ) . We also prove that ifθ − 1( E )is an M ‐set (respectively, M 1 ‐set), then E is an M ‐set (respectively, M 1 ‐set) and if Bim ( I ⊥ ) is the masa bimodule generated by the annihilator of the ideal I in V N ( G ) , then there exists an ideal J such that G ( Bim ( I ⊥ ) ) = Bim ( J ⊥ ) . If this ideal J is an ideal of multiplicity, then I is an ideal of multiplicity. In caseθ ∗ ( μ )is a Haar measure for θ ( G ) , we show that J is equal to the idealρ ∗ ( I )generated by ρ ( I ) , where ρ ( u ) = u ∘ θ ,∀ u ∈ I .