## Annals of Functional Analysis

### 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 $\phi\in\widehat{G/H}$, the right $\phi$-amenability of $L^{1}(G/H,\mu)$ and the right $\phi$-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 $\phi$-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.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 4 (2016), 564-572.

Dates
Accepted: 20 March 2016
First available in Project Euclid: 31 August 2016

https://projecteuclid.org/euclid.afa/1472659942

Digital Object Identifier
doi:10.1215/20088752-3661431

Mathematical Reviews number (MathSciNet)
MR3543148

Zentralblatt MATH identifier
06621461

#### Citation

Ramezanpour, M.; Tavallaei, N.; Olfatian Gillan, B. Character amenability and contractibility of some Banach algebras on left coset spaces. Ann. Funct. Anal. 7 (2016), no. 4, 564--572. doi:10.1215/20088752-3661431. https://projecteuclid.org/euclid.afa/1472659942

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