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...

Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if...