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Algebraic characterizations of locally compact groups

Part of: Special classes of linear operators Maps and general types of spaces defined by maps Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Juan J. Font  and
Salvador Hernández
Juan J. Font
Affiliation:
Departamento de Matemáticas Universidad Jaume I Campus Penyeta, E-12071 CastellónSpain e-mail: font@mat.uji.es, hernande@mat.uji.es fax: 34-64-345847
Salvador Hernández
Affiliation:
Departamento de Matemáticas Universidad Jaume I Campus Penyeta, E-12071 CastellónSpain e-mail: font@mat.uji.es, hernande@mat.uji.es fax: 34-64-345847
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Abstract

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Let G1, G2 be locally compact real-compact spaces. A linear map T defined from C(G1) into C(G2) is said to be separating or disjointness preserving if f = g ≡ 0 implies Tf = Tg ≡ 0 f or all f, g ∈ C(G1). In this paper we prove that both a separating map which preserves non-vanishing functions and a separating bijection which satisfies condition (M) (see Definition 4) are automatically continuous and can be written as weighted composition maps. We also study the effect of separating surjections (respectively injections) on the underlying spaces G1 and G2.

Next we apply the above results to give an algebraic characterization of locally compact Abelian groups, similar to the one given in [7] for compact Abelian groups in the presence of ring isomorphisms.

Finally, locally compact (not necessarily Abelian) groups are considered. We provide a sharpening of a result of Edwards and study the effect of onto (respectively injective) weighted composition maps on the groups G1 and G2.

MSC classification

Secondary: 47B38: Operators on function spaces (general) 43A20: $L^1$-algebras on groups, semigroups, etc. 54C40: Algebraic properties of function spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 3 , June 1997 , pp. 405 - 420
Copyright
Copyright © Australian Mathematical Society 1997

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