Strict topology as a mixed topology on Lebesgue spaces

STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

Saeid Maghsoudi

• Department of Mathematics, University of Zanjan, Zanjan, 45195-313, Iran

Rasoul Nasr-Isfahani

• Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 841546-83111, Iran

• School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box: 19395-5746, Tehran, Iran

Abstract

Let X be a locally compact space, and M∞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 M¹(X,ι) such that the strong dual of (M¹(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 algebras; Lebesgue spaces; locally compact group; locally convex topology; mixed topology; Radon measure; strict topology.

MSC:  primary 46A03; 46A70; secondary 46H05; 43A20.