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Amenability and invariant subspaces

Published online by Cambridge University Press:  09 April 2009

Anthony To-Ming Lau
Anthony To-Ming Lau
Affiliation:
Department of Mathematics, University of Alberta, Edmonton 7, Alberta, Canada
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Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 2 , September 1974 , pp. 200 - 204
Copyright
Copyright © Australian Mathematical Society 1974

References

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