We investigate conditions for the extendibility of continuous algebra homomorphisms ϕ from the Fourier algebra A(F ) of a locally compact group F to the Fourier-Stieltjes algebra B(G) of a locally compact group G to maps between the corresponding L∞ algebras which are weak* continuous. When ϕ is completely bounded and F is amenable, it is induced by a piecewise affine map α : Y → F where Y ⊆ G. We show that extendibility of ϕ is equivalent to α being an open map. We also study the dual problem for contractive homomorphismsϕ : L1 (F ) → M (G). We show that ϕ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map θ (Greenleaf [1965], Stokke [2011]) is a proper map.

Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when α 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 (α×α)−1 preserves sets of compact operator synthesis, and conversely when α 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.

We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms in the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. We prove that the existence of idempotents of arbitrarily large norm in B(G) implies the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for every locally compact group H. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for some amenable locally compact group H implies the existence of idempotents of arbitrarily large norm in B(G).

Bimodules over vN(G), harmonic operators and  the non-commutative Poisson boundary.

Starting with a left ideal J of L1(G) we consider its annihilator J ⊥ in L ∞ (G) and the generated VN(G)-bimodule in B(L 2 (G)), Bim(J ⊥ ). We prove that Bim(J ⊥ ) = (Ran J) ⊥ when G is weakly amenable discrete, compact or abelian, where Ran J is a suitable saturation of J in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the VN(G)-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski – Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by G.