We study certain commutative regular semisimple Banach algebras which we call hyperTauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A p (G) of a locally compact group G for p ∈ (1, ∞). We show that A p (G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A p (G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A p (G) is operator weakly amenable.The idea of this paper was initially motivated by the study of local derivations from commutative regular semisimple Banach algebras. Let A be a Banach algebra, and let X be a Banach A-bimodule. An operator D : A → X is a local derivation if for each a ∈ A, there is a derivations D a : A → X such that D(a) = D a (a).
Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ be a
2-cocycle, and let $\Phi$ be a Young function. In this paper, we consider the
Orlicz space $L^\Phi(G)$ and investigate its algebraic property under the
twisted convolution $\circledast$ coming from $\Omega$. We find sufficient
conditions under which $(L^\Phi(G),\circledast)$ becomes a Banach algebra or a
Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we
study its harmonic analysis properties, such as symmetry, existence of
functional calculus, regularity, and having Wiener property, mostly for the
case when $G$ is a compactly generated group of polynomial growth. We apply our
methods to several important classes of polynomial as well as subexponential
weights and demonstrate that our results could be applied to variety of cases
Abstract. We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L 1 (G), and the Fourier algebra, A(G), of a locally compact group G.
Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a * -representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π : A(G) → B(H ) is similar to a * -representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a * -representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].
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