### A characterization of groups with the one-sided Wiener property.

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Let $G$ be a locally compact group. We continue our work [A. Ghaffari: $\Gamma $-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of $\Gamma $-amenability of a locally compact group $G$ defined with respect to a closed subgroup $\Gamma $ of $G\times G$. In this paper, among other things, we introduce and study a closed subspace ${A}_{\Gamma}^{p}\left(G\right)$ of ${L}^{\infty}\left(\Gamma \right)$ and then characterize the $\Gamma $-amenability of $G$ using ${A}_{\Gamma}^{p}\left(G\right)$. Various necessary and sufficient conditions are found for a locally compact group to possess...

In the current work, a new notion of $n$-weak amenability of Banach algebras using homomorphisms, namely $(\varphi ,\psi )$-$n$-weak amenability is introduced. Among many other things, some relations between $(\varphi ,\psi )$-$n$-weak amenability of a Banach algebra $\mathcal{A}$ and ${M}_{m}\left(\mathcal{A}\right)$, the Banach algebra of $m\times m$ matrices with entries from $\mathcal{A}$, are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra ${L}^{1}\left(G\right)$ is ($\varphi ,\psi $)-$n$-weakly amenable for any...

For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let ${C}_{c}\left(G\right)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $({C}_{c}\left(G\right),{\epsilon}_{G})$ has property (FH) if and only if G has property (T). On...

We study the relationship between the classical invariance properties of amenable locally compact groups G and the approximate diagonals possessed by their associated group algebras L¹(G). From the existence of a weak form of approximate diagonal for L¹(G) we provide a direct proof that G is amenable. Conversely, we give a formula for constructing a strong form of approximate diagonal for any amenable locally compact group. In particular we have a new proof of Johnson's Theorem: A locally compact...

Let 𝓐 be a Banach algebra and let ϕ be a nonzero character on 𝓐. We give some necessary and sufficient conditions for the left ϕ-contractibility of 𝓐 as well as several hereditary properties. We also study relations between homological properties of some Banach left 𝓐-modules, the left ϕ-contractibility and the right ϕ-amenability of 𝓐. Finally, we characterize the left character contractibility of various Banach algebras related to locally compact groups.

Let G be a locally compact group with left Haar measure μ, and let L1(G) be the convolution Banach algebra of integrable functions on G with respect to μ. In this paper we are concerned with the investigation of the structure of G in terms of analytic semigroups in L1(G).