Herz Algebras Research Articles

A Characterization of Multipliers of the Herz Algebra

For the characterization of multipliers of Lp(Rd) or more generally, of Lp(G) for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra Ap(G) plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of Ap(G). In fact, we demonstrate that it coincides with the space of multipliers of Lp(G),∥·∥p. Given a multiplier T of (Ap(G),∥·∥Ap(G)) and using the embedding (Ap(G),∥·∥Ap(G))↪C0(G),∥·∥∞, the linear functional f↦[T(f)(0)] is bounded, and T can be written as a moving average for some element in the dual PMp(G) of (Ap(G),∥·∥Ap(G)). A key step for this identification is another elementary fact: showing that the multipliers from Lp(G),∥·∥p to C0(G),∥·∥∞ are exactly the convolution operators with kernels in Lq(G),∥·∥q for 1<p<∞ and 1/p+1/q=1. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra S0(G),∥·∥S0, and the fact that multipliers T from S0(G) to S0′(G) are convolution operators of the form T:f↦σ∗f for some uniquely determined σ∈S0′. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of S0′(G),∥·∥S0′. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications.

Weak⁎-continuity of invariant means on spaces of matrix coefficients

With every locally compact group G, one can associate several interesting bi-invariant subspaces X(G) of the weakly almost periodic functions WAP(G) on G, each of which captures parts of the representation theory of G. Under certain natural assumptions, such a space X(G) carries a unique invariant mean and has a natural predual, and we view the weak⁎-continuity of this mean as a rigidity property of G. Important examples of such spaces X(G), which we study explicitly, are the algebra McbAp(G) of p-completely bounded multipliers of the Figà-Talamanca–Herz algebra Ap(G) and the p-Fourier–Stieltjes algebra Bp(G). In the setting of connected Lie groups G, we relate the weak⁎-continuity of the mean on these spaces to structural properties of G. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.

On the extreme non‐Arens regularity of Banach algebras

As is well-know, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper we show that $\ell^1$-bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai's) and bounded nets converging to invariance (TI-nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non-Arens regular algebras. These criteria can be applied to the main algebras in harmonic analysis such as the group algebra, the measure algebra, the semigroup algebra (with certain weights) and the Fourier algebra. In this paper, we apply our criteria to the Lebesgue-Fourier algebra, the 1-Segal Fourier algebra and the Fig\`a-Talamanca Herz algebra.

Weakly almost periodic and uniformly continuous functionals on the Orlicz Figà-Talamanca Herz algebras

In this paper we study weakly almost periodic and uniformly continuous functionals on the Orlicz Figà-Talamanca Herz algebras associated to a locally compact group. We show that a unique invariant mean exists on the space of weakly almost periodic functionals. We also characterise discrete groups in terms of the inclusion of the space of uniformly continuous functionals inside the space of weakly almost periodic functionals.

AMENABILITY AND ORLICZ FIGÀ-TALAMANCA HERZ ALGEBRAS

AbstractIn this paper, we characterize amenability of locally compact groups in terms of the properties of Orlicz Figà-Talamanca Herz algebras.

Fig$$\grave{\hbox {a}}$$–Talamanca–Herz–Orlicz Algebras and Convoluters of Orlicz Spaces

Let G be a locally compact group and $$(\Phi , \Psi )$$ be a complementary pair of N-functions. In this paper, the Fig $$\grave{\hbox {a}}$$ –Talamanca–Herz algebras and the space of p-pseudomeasures on G are extended to Orlicz spaces. Indeed, the Banach algebra of $$\Phi $$ -pseudomeasures $${\mathop {\mathrm{PM}}}_{\Phi }(G)$$ and the Fig $$\grave{\hbox {a}}$$ –Talamanca–Herz–Orlicz algebras $${\mathop {\mathrm{A}}}_{\Phi }(G)$$ are defined. Then, it is shown that $${\mathop {\mathrm{A}}}_{\Phi }(G)^*={\mathop {\mathrm{PM}}}_{\Psi }(G)$$ . Furthermore, we characterize $$\mathop {\mathrm{Cv}}_{\Phi }(G)$$ , the space of $$\Phi $$ -convoluters, in terms of right translation invariant operators on $$M^{\Phi }(G)$$ . Then when G is amenable, we show that $$\mathop {\mathrm{Cv}}_{\Phi }(G)$$ , is equal to $$\mathop {\mathrm{PM}}_{\Phi }(G)$$ , a generalization of the classical p-version. Finally, we study $$B_{\Phi }(G)$$ , the space of pointwise multipliers of $$\mathop {\mathrm{A}}_{\Phi }(G)$$ when G is amenable.

On $${{\varvec{p}}}$$-completely contractive homomorphisms of Figà-Talamanca–Herz algebras

For a locally compact group G and $$1<p<\infty $$, let $$A_p(G)$$ be the Figa-Talamanca–Herz algebras, which include in particular the Fourier algebra of G, A(G) ($$p=2$$). In this note, we show that it is possible to describe an algebra homomorphism $$\Phi : A_p(G)\rightarrow A_p(H)$$ in terms of proper affine maps between the groups, under conditions that involve norm bounds on the second amplification of the homomorphism. This improves results by the first and second author (J Math Anal Appl 419(1):273–284, 2014; J Oper Theory 78:227–243, 2017). In particular, our result applies to p-completely bounded idempotent multipliers. As a corollary, we obtain that Figa-Talamanca–Herz algebras determine the underlying group up to p-completely contractive algebra isomorphisms.

Survey on the Figà–Talamanca Herz algebra

This paper presents a self contained approach to the theory of convolution operators on locally compact groups (both commutative and non commutative) based on the use of the Figà–Talamanca Herz algebras. The case of finite groups is also considered.

Orlicz Figà – Talamanca Herz algebras and invariant means

The aim of this paper is to initiate a systematic study of the Orlicz Figà – Talamanca Herz algebras on locally compact groups. We also introduce and study invariant means on the dual of these algebras.

Orthogonally additive and multiplicative maps between Figá-Talamanca–Herz algebras

We study the structure of an n-homogeneous polynomial P:Ap(G1)→Ap(G2) between Figá-Talamanca–Herz algebras. Among other results, we show that there exist a proper affine map α:Y⊂G2→G1 defined in an open coset Y of G2 such thatP(f)(y)=f(α(y))n,∀f∈Ap(G1),y∈Y when the locally compact groups G1,G2 are amenable and P is p-completely contractive, orthogonally additive and multiplicative, and preserves a right identity of Ap(G1)⁎⁎.

P-Analog of the Semigroup Fourier-Steiltjes Algebras

In this paper we dene the p-analog of the restricted rep- resentations and the p-analog of the Fourier{Stieltjes algebras on inverse semigroups. Also we improve some results about Herz algebras on Cliord semigroups and we give a necessary and sucient condition for amenabil- ity of these algebras on Cliord semigroups.

Peripherally multiplicative maps between Fig\`{a}--Talamanca--Herz algebras

Peripherally multiplicative maps between Fig\`{a}--Talamanca--Herz algebras

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G and 1<p<∞ let Ap(G) be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G, A(G) (p=2). It is shown that for any amenable group H, a proper affine map α:Y⊂H→G induces a p-completely contractive algebra homomorphism ϕα:Ap(G)→Ap(H) by setting ϕα(u)=u∘α on Y and ϕα(u)=0 off of Y. Moreover, we show that if both G and H are amenable then any p-completely contractive algebra homomorphism ϕ:Ap(G)→Ap(H) is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.

On the density of the Fourier–Stieltjes algebra

Let G be a locally compact group and 1<p<∞. Let Ap(G) denote the Figà-Talamanca–Herz algebra and MAp(G) its pointwise multiplier algebra. In this paper, we shall introduce two new spaces of multipliers of Ap(G), defined as coefficient functions of certain representations of G on some interpolation couples. Then we shall show that, in the amenable case, the Fourier–Stieltjes algebra B(G) is dense in MAp(G), and that MAp(G), as an ordered space, is a complete invariant for G.

Ideals with bounded approximate identities in the Fourier algebras on homogeneous spaces

In this paper, the closed ideals with bounded approximate identities in the Fourier algebra of an amenable homogeneous space is characterized in terms of the coset ring of the corresponding group. Some of these results are also extended to the case of Figà–Talamanca–Herz algebra. The closure of the Fourier algebra A(G/K) in the cb-multiplier norm is also considered and we prove some results on spectral synthesis. We also derive some results on the ideals with bounded approximate identity.

Figa-Talamanca–Herz algebras for restricted inverse semigroups and Clifford semigroups

We develop the Figa-Talamanca–Herz algebras and the space of p-pseudomeasures to inverse semigroups with restricted semigroup algebras. Let 1<p,q<∞ be such that 1p+1q=1. We define the Banach algebra of p-pseudomeasures PMp(S) and the Figa-Talamanca–Herz algebras Aq(S). Then we show that Aq(S)∗=PMp(S). We characterize PMp(S) and Aq(S) for a Clifford semigroup, in the sense of p-pseudomeasures and Figa-Talamanca–Herz algebras of maximal subgroups of S, respectively. We also show that the character space of Aq(S) is equal to S for a Clifford semigroup S. As an example of these Banach algebras and restricted semigroup algebras, we discuss uniformly locally finite inverse semigroups.

Representing Multipliers of the Fourier Algebra on Non-Commutative Lp Spaces

Abstract We show that the multiplier algebra of the Fourier algebra on a locally compact group G can be isometrically represented on a direct sum on non-commutative Lp spaces associated with the right von Neumann algebra of G. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative Lp spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative Lp spaces, say . It is shown that is isometric to L1(G), generalising the abelian situation.

A note on weighted A_p(G)-modules

Let G be a locally compact abelian group and w be a weight function on G. In this paper, we show that the space A_{p,w}(G) is a Banach module over the Figà-Talamanca Herz algebra A_p(G) and study the multiplier space from A_p(G) to A_{p,w}(G).

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1 < p < ∞ . We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on L p -spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figà–Talamanca–Herz algebra A p ( G ) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗-algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p-completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.

Cohen–Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca–Herz algebras

Let G be a locally compact group, and let R ( G ) denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen–Host idempotent theorem asserts that a set lies in R ( G ) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen–Host type theorems to the study of the Figà-Talamanca–Herz algebras A p ( G ) with p ∈ ( 1 , ∞ ) . For arbitrary G, we characterize those closed ideals of A p ( G ) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that A p ( G ) is 1-amenable for some—and, equivalently, for all— p ∈ ( 1 , ∞ ) : these are precisely the abelian groups.