Strict topology as a mixed topology on Lebesgue spaces

Authors

  • Saeid Maghsoudi

Keywords:

Lebesgue spaces, mixed topology, group algebras, Radon measure

Abstract

Let $X$ be a locally compact space, and $\Lz$ be the space of all essentially bounded $\iota$-measurable functions $f$ on $X$ vanishing at infinity. We introduce and study a locally convex topology $\beta^1(X,\iota)$ on the Lebesgue space $\Lo$ such that the strong dual of $(\Lo,\beta^1(X,\iota))$ can be identified with $(\Lz,\|.\|_\infty)$. Next by showing that $\beta^1(X,\iota)$ topology can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove that $L_1(G)$, the group algebras of a locally compact Hausdorff topological group $G$, equipped with the convolution multiplication is a complete semi-topological algebra under $\beta^1(G)$ topology. doi:10.1017/S0004972711002589

Published

2011-10-18

Issue

Vol. 84 No. 3 (2011)

Section

Articles
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