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 µ be a strongly quasi-invariant Radon measure on the homogeneous space G/H. In this article, we show that every element of G/H, the character space of G/H, determines a nonzero multiplicative linear functional on L 1 (G/H, µ). Using this, we prove that for all φ ∈ G/H, the right φ-amenability of L 1 (G/H, µ) and the right φ-amenability of M (G/H) are both equivalent to the amenability of G. Also, we show that L 1 (G/H, µ), as well as M (G/H), is right φ-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.