### 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations

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Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality $\mu \u0332\left(AB\right)\ge min\left(\mu \u0332\right(A)+\mu \u0332(B),\mu (G\left)\right)$ for unimodular G.

In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

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...

Answering a question of Pisier, posed in [10], we construct an L-set which is not a finite union of translates of free sets.