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 $𝒜$ and $M_m\left(𝒜\right)$, the Banach algebra of $m\times m$ matrices with entries from $𝒜$, 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...