Let $G$ be a locally compact group, and let ${\cal R}(G)$ 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 ${\cal R}(G)$ 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 $p \in (1,\infty)$. 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 -- $p \in (1,\infty)$: these are precisely the abelian groups.
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 functions inside the space of weakly almost periodic functionals.
Let G be a locally compact group. In this note, we characterise non-degenerate *-representations of A_Φ(G) and B_Φ(G). We also study spectral subspaces associated to a non-degenerate Banach space representation of A_Φ(G).
For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling-Figa-Talamanca-Herz algebras $A_p(G,ω)$. For $p=2$ and abelian $G$, these are precisely the Beurling algebras on the dual group $\hat{G}$. For $p =2$ and compact $G$, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group $G$ is amenable if and only if one - and, eq…
We introduce a noncommutative analogue of the Figá-Talamanca-Herz algebra $A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of completely bounded Schur multipliers on Schatten space $S_p$. We determine the isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded Schur multipliers on Schatten space $S_p$ is the closure in the weak operator topology of the span of isometric multipliers.
Let $G$ be a locally compact group and $p,q\in \mathbb{R}$ with $p>1$ $p\not=2$ and $q$ between $2$ and $p$ (if $p<2$ then $p<q<2$, if $p>2$ then $2<q<p.$) The main result of the paper is that $A_q(G)$ multiplies $A_p(G)$, more precisely we show that the Banach algebra $A_p(G)$is a Banach module on $A_q(G).$
We consider the minimal boundedly-translation-invariant Segal algebra $S_0^p(G)$ in the Figà-Talamanca--Herz algebra $A_p(G)$ of a locally compact group $G$. In the case that $p=2$ and $G$ is abelian this is the classical Segal algebra of Feichtinger. Hence we call this the Feichtinger--Figà-Talamanca--Herz Segal algebra of $G$. Remarkably, this space is also a Segal algebra in $L^1(G)$ and is, in fact, the minimal such algebra which is closed under pointwise multiplication by $\apg$. Even for $p=2$, this result is new for non-abelian $G$. We place a $p$-operator space structure on $S_0^p(G)$,…
Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p(G)$ for $p \in [1,\infty]$ introduced by G. Pisier to define operator space analogues $OA_p(G)$ of the classical Figa-Talamanca-Herz algebras $A_p(G)$. If $p \in (1,\infty)$ is arbitrary, then $A_p(G) \subset OA_p(G)$ such that the inclusion is a contraction; if p = 2, then $OA_2(G) \cong A(G)$ as Banachspaces spaces, but not necessarily as operator spaces. We show that $OA_p(G)$ is a completely contractive Banach algebra for each $p \in (1,\infty)$, and that $OA_q(G) \subset OA_p(G)$ completely …
Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group $G$ and $p,p' \in (1,\infty)$ with $\frac{1}{p} + \frac{1}{p'} = 1$, we use the operator space structure on $CB(COL(L^{p'}(G)))$ to equip the Figa-Talamanca-Herz algebra $A_p(G)$ with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for $p \leq q \leq 2$ or $2 \leq q \leq p$ and amenable $G$, the cano…
It is shown that a locally compact group $G$ is amenable if and only if some certain closed ideals of the Figà-Talamanca-Herz algebra $A_{p}(G)$ admit bounded $Δ$-weak approximate identities. Also, similar results are obtained for the function algebras $\mathfrak{L}A(G)$ and $C_{0}^{w}(G)$.
Let $A$ be a Banach algebra with a non-empty character space. We say that a bounded net $\{e_α\}$ in $A$ is a bounded $Δ$-weak approximate identity for $A$ if, for each $a\in A$ and compact subset $K$ of $Δ(A)$, $||\widehat{e_αa}-\widehat{a}||_{K}=\sup_{ϕ\in K}|ϕ(e_αa)-ϕ(a)|\rightarrow 0$. For each $1<p<\infty$, we prove that the Figa-Talamanca Herz algebra, $A_{p}(G)$ has a bounded $Δ$-weak approximate identity if and only if $G$ is an amenable group. Also we give a sufficient condition for amenability of group $G$.
Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally compact group}} $G$, thus $A_2(G)$ {\it{is the Fourier Algebra of $G$. If $G$ is commutative then $A_2(G)=L^1(\hat{G})^\wedge$. Let $A_p^r(G)=A_p\cap L^r(G)$ with norm $\|u\|_{A_p^r}=\| u\|_{A_p}+\| u\|_{L^r}$.We investigate for which $p$, $r$, and $G$ do the Banach algebras $A_p^r(G)$ {\it{have the Banach space geometric properties: The Radon-Nikodym Property (RNP), the Schur Property (SP) or the Dunford-Pettis Property (DPP). The results are new even if $G=R$ (the real line) or $G=Z$ (the additive integers).
Let $G$ be a locally compact group and $(Φ, Ψ)$ be a complementary pair of $N$-functions. In this paper, using the powerful tool of porosity, it is proved that when $G$ is an amenable group, then the Figà-Talamanca-Herz-Orlicz algebra ${\A}_Φ(G)$ is a Banach algebra under convolution product if and only if $G$ is compact. Then it is shown that ${\A}_Φ(G)$ is a Segal algebra, and as a consequence, the amenability of ${\A}_Φ(G)$ and the existence of a bounded approximate identity for ${\A}_Φ(G)$ under the convolution product is discussed. Furthermore, it is shown that for a compact abelian group…
Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally compact group $G$, thus $A_2(G)$ is the Fourier Algebra of $G$. If $G$ is commutative then $A_2(G)=L^1(\hat{G}){\hat{}}$. Let $A^r_p(G)=A_p\cap L^r(G)$ with norm $||u||_{A_p^r}=||u||_{A_p}+||u||_{L^r}$. We investigate a property which insures not only existence of solutions to optimization problems but moreover, facility in testing that an algorithm converges to such solutions namely the RNP. Theorem(a): If $G$ is weakly amenable then $A_p^r$ is a dual Banach space with RNP if $1\leq r\leq p'$. This does not hold if $G=S…
For a locally compact group $G$ and $p \in (1,\infty)$, we define $B_p(G)$ to be the space of all coefficient functions of isometric representations of $G$ on quotients of subspaces of $L_p$ spaces. For $p =2$, this is the usual Fourier--Stieltjes algebra. We show that $B_p(G)$ is a commutative Banach algebra that contractively (isometrically, if $G$ is amenable) contains the Figà-Talamanca--Herz algebra $A_p(G)$. If $2 \leq q \leq p$ or $p \leq q \leq 2$, we have a contractive inclusion $B_q(G) \subset B_p(G)$. We also show that $B_p(G)$ embeds contractively into the multiplier algebra of $A_…
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 $L^p$ spaces associated to the right von Neumann algebra of $G$. 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 non-commutative $L^p$ spaces we construct, and show that they are completely isometric to those considered recently by Forrest, Lee and Samei; we improve a result about module homomo…
With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ 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 $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca…
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 beh…
On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $λ\in \mathbb{C}$. This is possible whenever $λ$ is in the resolvent set of $P$ as a self-adjoint operator on a suitable $\ell^2$-space and the on-diagonal elements of the resolvent ("Green function") do not vanish at $λ$. We show that when $P$ is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition h…
