Character amenability and contractibility of some Banach algebras on left coset spaces

Abstract

Let $H$ be a compact subgroup of a locally compact group $G$, and let $\mu$ be a strongly quasi-invariant Radon measure on the homogeneous space $G/H$. In this article, we show that every element of $\widehat{G/H}$, the character space of $G/H$, determines a nonzero multiplicative linear functional on $L^1(G/H,\mu )$. Using this, we prove that for all $\varphi \in \widehat{G/H}$, the right $\varphi$-amenability of $L^1(G/H,\mu )$ and the right $\varphi$-amenability of $M(G/H)$ are both equivalent to the amenability of $G$. Also, we show that $L^1(G/H,\mu )$, as well as $M(G/H)$, is right $\varphi$-contractible if and only if $G$ is compact. In particular, when $H$ is the trivial subgroup, we obtain the known results on group algebras and measure algebras.