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Let $X$ be a hypergroup. In this paper, we define a locally convex topology $\beta$ on $L\left(X\right)$ such that ${(L\left(X\right),\beta )}^{*}$ with the strong topology can be identified with a Banach subspace of $L{\left(X\right)}^{*}$. We prove that if $X$ has a Haar measure, then the dual to this subspace is $L_C{\left(X\right)}^{**}{=cl\{F\in L\left(X\right)}^{**};F$ has compact carrier}. Moreover, we study the operators on $L{\left(X\right)}^{*}$ and $L_0^{\infty }\left(X\right)$ which commute with translations and convolutions. We prove, among other things, that if $wap\left(L\right(X\left)\right)$ is left stationary, then there is a weakly compact operator $T$ on $L{\left(X\right)}^{*}$ which commutes with convolutions if and...

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...