The space of vector‐valued integrable functions under certain locally convex topologies

Abstract

Let E be a Banach space, Ω a locally compact space, and μ a positive Radon measure on Ω. In this paper, through extending to Lebesgue‐Bochner spaces, we show that the topology β1 introduced by Singh is a type of strict topology. We then investigate various properties of this locally convex topology. In particular, we show that the strong dual of L1(μ, E) can be identified with a Banach space. We also show that the topology β1 is a metrizable, barrelled or bornological space if and only if Ω is compact. Finally, we consider the generalized group algebra \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^1(G, \mathbf {A})$\end{document} under certain natural locally convex topologies. As an application of our results, we prove that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^1(G,\mathbf {A})$\end{document} under the topology β1 is a complete semi‐topological algebra.

Hypergroup Algebras as Topological Algebras

“…This topology has been introduced and denoted by β 1 in [18] and [3] for locally compact groups and hypergroups, respectively. For a similar recent study in other contexts, see [11][12][13].…”

“…Throughout this work, let K denote a locally compact hypergroup with a fixed left Haar measure m. We begin with the following result in which we collect some properties of the topology β 1 that we believe are interesting (for the proofs in a more general setting, we refer the reader to [11]). Proposition 2.1.…”

“…Recently, it was shown [11] that the β 1 -topology can be viewed as a type of generalised strict topology in the sense of Sentilles and Taylor [15]. To this end, we consider the Banach space L 1 (K) as a Banach left B 0 (K)-module, where the module action is the natural pointwise product of functions.…”

“…Here B 0 (K) stands for the Banach algebra of all bounded Borel measurable functions on K vanishing at infinity under a pointwise product of functions. Then L 1 (K) is an essential B 0 (K)-module (see [11] for more details).…”

A strict topology on Orlicz spaces

“…We start with the following key lemma that gives a neighborhood base at zero for the topology . It is an extension of Lemma 3.1 in for the more general setting of Orlicz spaces. Lemma Let Ω be a locally compact space and μ a Radon measure on Ω.…”