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Let $S$ be an inverse semigroup with the set of idempotents $E$ and $S/\approx$ be an appropriate group homomorphic image of $S$. In this paper we find a one-to-one correspondence between two cohomology groups of the group algebra ${\ell }^1\left(S\right)$ and the semigroup algebra ${\ell }^1(S/\approx )$ with coefficients in the same space. As a consequence, we prove that $S$ is amenable if and only if $S/\approx$ is amenable. This could be considered as the same result of Duncan and Namioka [5] with another method which asserts that the inverse semigroup $S$ is amenable...

In this paper, we introduce multi-Jensen-cubic mappings and unify the system of functional equations defining the multi-Jensen-cubic mapping to a single equation. Applying a fixed point theorem, we establish the generalized Hyers-Ulam stability of multi-Jensen-cubic mappings. As a known outcome, we show that every approximate multi-Jensen-cubic mapping can be multi-Jensen-cubic.

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 completely contractive Banach algebra $B$, we find conditions under which the completely bounded multiplier algebra ${ℳ}_{cb}\left(B\right)$ is a dual Banach algebra and the operator amenability of $B$ is equivalent to the operator Connes-amenability of ${ℳ}_{cb}\left(B\right)$. We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.

We find some relations between module biprojectivity and module biflatness of Banach algebras $𝒜$ and $ℬ$ and their projective tensor product $𝒜\widehat{\otimes }ℬ$. For some semigroups $S$, we study module biprojectivity and module biflatness of semigroup algebras $l^1\left(S\right)$.

Let A be a Banach algebra, E be a Banach A-bimodule and Δ E → A be a bounded Banach A-bimodule homomorphism. It is shown that under some mild conditions, the weakΔ''-amenability of E'' (as an A''-bimodule) necessitates weak Δ-amenability of E (as an A-bimodule). Some examples of weak-amenable Banach modules are provided as well.

In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form [...] f(3x+y)−5f(2x+y)+f(2x−y)+10f(x+y)−5f(x−y)=10f(y)+4f(2x)−8f(x) $$f\left(3x+y\right)-5f\left(2x+y\right)+f\left(2x-y\right)+10f\left(x+y\right)-5f\left(x-y\right)=10f\left(y\right)+4f\left(2x\right)-8f\left(x\right)$$ in the set of real numbers.

The generalized notion of weak amenability, namely $(\varphi ,\psi )$-weak amenability, where $\varphi ,\psi$ are continuous homomorphisms on a Banach algebra $𝒜$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi ,\psi )$-weak amenability on the measure algebra $M\left(G\right)$, the group algebra $L^1\left(G\right)$ and the Segal algebra $S^1\left(G\right)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi ,\psi )$-weak amenability of a special semigroup algebra is shown as well.