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Resumen:
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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 ? on a subspace X of B(S,E) , containing the constant functions, is a linear mapping of X into E such that ?(f) belongs to the closed convex hull of f(S) . Moreover. if X is left [right] invariant, ? is left [right] invariant provided ?( s f)=?(f)[?(f s )=?(f)] , for every f?X and s?S . If X is invariant and ? is left and right invariant, then ? 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.
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