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Every homomorphism $\varphi: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $\alpha: Y \rightarrow \Delta(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $\Delta(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $\alpha$ for which $\varphi=j_\alpha: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractive/completely bounded homomorphisms $\varphi: B(G) \rightarrow B(H)$ when $G$ is a Euclidean- or $p$-adic-motion group. In these cases, our description of the completely positive/completely contractive homomorphisms employs the notion of a "fusion map of a compatible system of homomorphisms/affine maps" and is quite different from the Fourier algebra situation.

Using the recently developed notion of a Herz--Schur multiplier of a C*-dynamical system we introduce weak amenability of C*- and W*-dynamical systems. As a special case we recover Haagerup's characterisation of weak amenability of a discrete group. We also consider a generalisation of the Fourier algebra to crossed products and study its multipliers., 18 pages

Let $G$ be a locally compact group. We show how complemented ideals in the Fourier algebra $A(G)$ of $G$ arise naturally from a class of thin sets known as Leinert sets. Moreover, we also present an explicit example of a closed ideal in $A(\mathbb{F}_{N})$, the free group on $N \ge 2$ generators, that is complemented in $A(\mathbb{F}_{N})$ but it is not completely complemented. Then by establishing an appropriate extension result for restriction algebras arising from Leinert sets, we show that any almost connected group $G$ for which every complemented ideal in $A(G)$ is also completely complemented must be amenable., Comment: 23 pages

For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling-Figa-Talamanca-Herz algebras $A_p(G,\omega)$. 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, equivalently, every - Beurling-Figa-Talamanca-Herz algebra $A_p(G,\omega)$ has a bounded approximate identity.

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In the first part of the paper, we present a short survey of the theory of multipliers, or double centralisers, of Banach algebras and completely contractive Banach algebras. Our approach is very algebraic: this is a deliberate attempt to separate essentially algebraic arguments from topological arguments. We concentrate upon the problem of how to extend module actions, and homomorphisms, from algebras to multiplier algebras. We then consider the special cases when we have a bounded approximate identity, and when our algebra is self-induced. In the second part of the paper, we mainly concentrate upon dual Banach algebras. We provide a simple criterion for when a multiplier algebra is a dual Banach algebra. This is applied to show that the multiplier algebra of the convolution algebra of a locally compact quantum group is always a dual Banach algebra. We also study this problem within the framework of abstract Pontryagin duality, and show that we construct the same weak$^*$-topology. We explore the notion of a Hopf convolution algebra, and show that in many cases, the use of the extended Haagerup tensor product can be replaced by a multiplier algebra., Comment: Numerous typos corrected, and a proof rectified (v3 was an incorrect upload of a previous version)

Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, L^1(G) and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisimple commutative Banach algebras. We wish to emphasize that the theory of operator spaces and completely bounded maps plays an indispensable role when studying A(G) and B(G). We also show some applications of amenability theory to problems of complemented ideals and homomorphisms., Comment: 19 pages, survey article, accepted for publication in Banach algebras 2009 conference proceedings

We continue our study of derivations, multipliers, weak amenability and Arens regularity of Segal algebras on locally compact groups. We also answer two questions on Arens regularity of the Lebesgue{Fourier algebra left open in our earlier work. In this paper we continue our study of derivations, multipliers, weak amen- ability and Arens regularity of Segal algebras and in particular we show that a symmetric Segal algebra S(G) is approximately weakly amenable if one only assumes that G is a SIN group or an amenable group. We show that if G is amenable and S(G) is a symmetric Segal algebra, then for every Banach L 1 (G)-bimodule X; continuous derivations from S(G) into X are approximately inner. Alternatively, continuous derivations from S(G) into X are precisely the ones dened by continuous double centralizers. In (9) we also showed that the Lebesgue{Fourier algebra of a compact (discrete) group G is Arens regular for the convolution (resp. pointwise) product. Here we prove the converse to these results, provided that G is unimodular in the rst case, and for all groups in the second case. Thus we answer two questions left open in (9). For a compact group G; we determine the spaceZ 1 (S(G);X) of continu- ous derivations from S(G) into X; where S(G) is one of the naturally arising

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 contractively for amenable $G$ if $1 < p \leq q \leq 2$ or $2 \leq q \leq p < \infty$. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in $OA_p(G)$ for one (or equivalently for all) $p \in (1,\infty)$.