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STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

Part of: Topological linear spaces and related structures Abstract harmonic analysis

Published online by Cambridge University Press:  06 September 2011

SAEID MAGHSOUDI  and
RASOUL NASR-ISFAHANI
SAEID MAGHSOUDI*
Affiliation:
Department of Mathematics, Zanjan University, Zanjan, Iran Research Institute for Fundamental Science, Tabriz, Iran (email: s_maghsodi@znu.ac.ir)
RASOUL NASR-ISFAHANI
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (email: isfahani@cc.iut.ac.ir)
*
✉For correspondence; e-mail: s_maghsodi@znu.ac.ir
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Abstract

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Let X be a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially bounded ι-measurable functions f on X vanishing at infinity. We introduce and study a locally convex topology β1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with . Next, by showing that β1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove that L1 (G) , the group algebra of a locally compact Hausdorff topological group G, equipped with the convolution multiplication is a complete semitopological algebra under the β1 (G) topology.

Keywords

group algebrasLebesgue spaceslocally compact grouplocally convex topologymixed topologyRadon measurestrict topology

MSC classification

Secondary: 46A03: General theory of locally convex spaces 46A70: Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.) 43A20: $L^1$-algebras on groups, semigroups, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 504 - 515
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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