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- Author or Editor: Ali Ghaffari x
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Abstract
Let S be a foundation locally compact topological semigroup, and let M a (S) be the space of all measures μ∊M(S) for which the maps x↦|μ|∗δ x and x↦|μ|∗δ x from S into M(S) are weakly continuous. The purpose of this article is to develop a notion of character amenability for semigroup algebras. The main results concern the χ-amenability of M a (S). We give necessary and sufficient conditions for the existence of a left χ-mean on M a (S)∗.
Suppose that A is either the Banach algebra L 1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L 1(G) and M(G)) such as the second dual, L 1(G)**, of L 1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.
Given a foundation locally compact Hausdorff topological semigroup S , we consider on M a ( S )* the τ c -topology, i.e. the weak topology under all right multipliers induced by measures in M a ( S ). For such an arbitrary S the τ c -topology is not weaker than the weak*-topology and not stronger than the norm topology on M a ( S )*. However, a further investigation shows that for compact S the norm topology and τ c -topology coincide on every norm bounded subset of M a ( S ). Among the other results we mention that except for discrete S the τ c -topology is always different from the norm-topology. Finally, we give some results about τ c -almost periodic functionals.
