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Character amenability and contractibility of some Banach algebras on left coset spaces. (English) Zbl 1443.43004
Summary: 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.
##### MSC:
 43A20 $$L^1$$-algebras on groups, semigroups, etc. 46H05 General theory of topological algebras 43A07 Means on groups, semigroups, etc.; amenable groups
##### Keywords:
Banach algebra; homogeneous space; character amenability
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