Means in locally convex spaces and semireflexivity. (Spanish: Medias en espacios localmente convexos y semi-reflexividad).

Abstract

Let S be a semigroup and let E be a locally convex topological vector space over the field of real numbers. Let B(S,E) be the linear space of mappings f:S→E such that f(S) is a bounded set of E . For every s∈S and every f∈B(S,E), s f[f s ] denotes the function from S into E defined by ( s f)(T)=f(s⋅t) for each t∈S [(f s )(t)=f(t⋅s) for each t∈S ]. A subspace X of B(S,E) is left [right] invariant if, for every f∈X and s∈S , s f[f s ] also belongs to X . The space X is invariant if it is both left and right invariant. The authors give the following definition of a mean: a mean μ on a subspace X of B(S,E) , containing the constant functions, is a linear mapping of X into E such that μ(f) belongs to the closed convex hull of f(S) . Moreover. if X is left [right] invariant, μ is left [right] invariant provided μ( s f)=μ(f)[μ(f s )=μ(f)] , for every f∈X and s∈S . If X is invariant and μ is left and right invariant, then μ is invariant.
The authors study the problem of the existence of invariant means on certain subspaces of X . For a larger class of semigroups they prove that if E is quasicomplete for the Mackey topology, a necessary and sufficient condition to ensure the existence of a invariant mean on B(S,E) is that E be semi-reflexive.