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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 |
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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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