Completely bounded homomorphisms of the Fourier algebra revisited

Assume that A ⁢ ( G ) A(G) and B ⁢ ( H ) B(H) are the Fourier and Fourier–Stieltjes algebras of locally compact groups 𝐺 and 𝐻, respectively. Ilie and Spronk have shown that continuous piecewise affine maps α : Y ⊆ H → G \alpha\colon Y\subseteq H\to G induce completely bounded homomorphisms Φ : A ⁢ ( G ) → B ⁢ ( H ) \Phi\colon A(G)\to B(H) and that, when 𝐺 is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure-theoretic ideas, following more closely the original ideas of Cohen.

Dual Convolution for the Affine Group of the Real Line

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $$L^2({{\mathbb {R}}}^\times , dt/ |t|)$$ L 2 ( R × , d t / | t | ) . In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $$L^p({{\mathbb {R}}}^\times , dt/ |t|)$$ L p ( R × , d t / | t | ) for $$p\in (1,2)\cup (2,\infty )$$ p ∈ ( 1 , 2 ) ∪ ( 2 , ∞ ) .

SOME HOMOLOGICAL PROPERTIES OF FOURIER ALGEBRAS ON HOMOGENEOUS SPACES

Let $ H $ be a compact subgroup of a locally compact group $ G $ . We first investigate some (operator) (co)homological properties of the Fourier algebra $A(G/H)$ of the homogeneous space $G/H$ such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that $ A(G/H) $ is operator approximately biprojective if and only if $ G/H $ is discrete. We also show that $A(G/H)^{**}$ is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on $A(G/H)$ .

Metric semi-groups of normal states determine F-algebras

Let A be an F-algebra and S(A*) be the metric semi-group of normal states of the W*-algebra A*. We show that S(A*) determines A, when A* is of type I. In particular, when A is either the Fourier algebra A(G) or the Fourier–Stieltjes algebra B(G) of a type I locally compact group G, the metric semi-group S(A*) determines G up to opposition. A remark for the general case will also be given.