Completely bounded homomorphisms of the Fourier algebras

Abstract

For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H ) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine mapan element of the open coset ring) induces a completely bounded homomorphism : A(G) → B(H ) by setting u = u • on Y and u = 0 off of Y. We show that if G is amenable then any completely bounded homomorphism : A(G) → B(H ) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of Cohen (Amer.

The spine of a Fourier-Stieltjes algebra

“…We note that condition (ii) is equivalent to (ii') ker η τ 1 τ is compact and G τ ∼ = G τ 1 / ker η τ 1 τ , homeomorphically. Indeed, proper maps are closed (see [25,Corollary 3.11]…”

“…Motivated by the papers of Inoue [26,27], in Section 4 we exploit the semi-lattice structure on our locally precompact non-quotient topologies to construct the spectrum G * of A * (G). Using G * we can extend our generalisation [25] of Cohen's homomorphism theorem [11] to study homomorphisms from A * (G) to B(H) for another locally compact group H. We are forced, however, to consider the natural operator space structures on A * (G) and B(H), and as well restrict ourselves to amenable G. Thus we obtain a characterisation of completely bounded homomorphisms from the spine A * (G) for an amenable group G, to B(H) for a general locally compact group H. Moreover, we show that A * (H) is exactly the subspace of B(H) which contains the images of all such homomorphisms. We also obtain the facts that if H 0 is an open subgroup of H, then A * (H)| H 0 = A * (H 0 ), and that A * (H) contains all of the idempotents from B(H).…”

Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras

“…Corollary 3.2 in particular shows that if G and H are locally compact groups and H is amenable, then every homomorphism from A(H) into B(G) extends to a homomorphism from B(H) = M (A(H)) into B(G). The reader should compare this with results on extensions of completely bounded homomorphisms in Section 3 of [15].…”

“…Of course, homomorphisms between Banach algebras have been studied for a long time and by numerous authors from various different aspects (see [5], [7], [11], [15] and [16], to mention just a few, and the extensive list of references in [6]). …”

“…When G is amenable, in Theorem 3.13 of [15] the range of a completely bounded homomorphism has been identified as the set L ϕ = {a ∈ I ϕ : for γ 1 , γ 2 ∈ E ϕ , ϕ * (γ 1 ) = ϕ * (γ 2 ) ⇒ a(γ 1 ) = a(γ 2 )}.…”