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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.
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46H05 General theory of topological algebras
43A07 Means on groups, semigroups, etc.; amenable groups
Full Text: DOI Euclid