Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T\otimes I{d}_X$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^p\left(G\right)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions...

L’espace $P{F}_p\left(G\right)$ des $p$-pseudofonctions sur un groupe localement compact $G$ est le complété de $L_1\left(G\right)$ pour la norme de convoluteur de $L_p\left(G\right)$. Dans le cas où le groupe $G$ est moyennable alors le banach dual à $P{F}_p\left(G\right)$ s’identifie avec une certaine algèbre $B_p\left(G\right)$ de fonctions continues sur $G$. L’algèbre $B_p\left(G\right)$ est déjà connue mais ici on montre que $B_p$ est un foncteur de groupes localement compacts. Pour $p=2$ alors $P{F}_2\left(G\right)$ est l’algèbre $C^{*}$ de $G$ dont le dual est $FS\left(G\right)$, l’algèbre de transformées de Fourier-Stieltjes. Donc, pour un groupe moyennable, élément...

We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{\nu }=sup{\left|\right|(e-x)}^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,max\left|x̂\right|\le \nu$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{\delta }=sup\left|\right|x^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,min\left|x̂\right|\ge \delta$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty }\left(A\right)$, introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty }\left(A\right)<1$, and this result is extended to more general nonlinear extremal problems...

A group acting on a measure space (X,β,λ) may or may not admit a cyclic vector in $L_{\infty }\left(X\right)$. This can occur when the acting group is as big as the group of all measure-preserving transformations. But it does not occur, even though there is no cardinality obstruction to it, for the regular action of a group on itself. The connection of cyclic vectors to the uniqueness of invariant means is also discussed.

Let G be a locally compact abelian group and let X be a translation invariant linear subspace of L¹(G). If G is noncompact, then there is at most one Banach space topology on X that makes translations on X continuous. In fact, the Banach space topology on X is determined just by a single nontrivial translation in the case where the dual group Ĝ is connected. For G compact we show that the problem of determining a Banach space topology on X by considering translation operators on X is closely related...