Amenability induced by amenable homomorphic images

Abstract

Let S and S' be semitopological semigroups and 0:S+S' a continuous homomorphism onto S'. Denote by ~:C(S')+C(S) the dual map f§ where C(S) and C(S') denote the Banach spaces of all bounded, continuous, realvalued functions on S and S' respectively. M.M. Day has shown that if F' is a translation invariant linear subspace of C(S'), then F' is left amenable if and only if the image space ~(F') is left amenable [3,p. 541]. The corresponding result for inverse images is easily seen to be false. More precisely, if F is a translation invariant linear subspace of C(S) then, while left amenability of F always implies that ~-I(F) is left amenable, trivial examples show that the converse implication generally fails to hold. In section 1 of this paper we give a necessary and sufficient condition ~ for the converse implication to hold, under the assumptions that 8 is an open mapping and F is left introverted. In section 2 we formulate and prove an analog of this result for the case of topological left amenability. Our main results may be viewed as generalizations of theorems of T. Mitchell and E. Granirer A.T. Lau on stationary semigroups. Applications are made to quotient groups, semidirect products, and to various concrete function spaces.