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

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Bombal Gordón, Fernando and Vera Boti, Gabriel (1973) Means in locally convex spaces and semireflexivity. (Spanish: Medias en espacios localmente convexos y semi-reflexividad). Collectanea mathematica, 24 (3). pp. 267-295. ISSN 0010-0757

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Official URL: http://www.collectanea.ub.edu/index.php/Collectanea/article/view/3416/4096




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.


Item Type:Article
Uncontrolled Keywords:Means on groups;Locally convex spaces;
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:17849
Deposited On:22 Jan 2013 12:43
Last Modified:08 May 2018 08:09

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