A mixed topology over the space of measures with continuous translations on a locally compact semigroup
type: Item
instance of: scholarly article
title: A mixed topology over the space of measures with continuous translations on a locally compact semigroup
zbMATH Open document ID: 1266.22004
published in: Semigroup Forum
publication date: 2013-03-21T00:00:00Z
review text: Throughout this review \(S\) will always be a locally compact semigroup. Let \(M(S)\) denote the Banach space of all bounded complex-valued regular Borel measures on \(S\) with the total variation norm. Write \(n(S)\) for the topology generated by the total variation norm on \(M(S)\). The space of all \(\mu \in M(S)\) for which the functions \(x \mapsto | \mu | (x^{-1}K)\) and \(x \mapsto | \mu | (Kx^{-1})\) are continuous on \(S\), for all compact subsets \(K\), is denoted by \(M_a(S)\). Let \(\mathcal{C}\) be the set of increasing sequences of compact subsets of \(S\) and let \(\mathcal{R}\) be the set of increasing sequences \((r_n)\) of real numbers in \((0, \infty)\) with \(r_n \rightarrow \infty\). For any \((C_n) \in \mathcal{C}\) and \((r_n) \in \mathcal{R}\), set \[ U( (C_n), (r_n)) = \{ \mu \in M_a(S) : | \mu |(C_n) \leq r_n \text{ for all } n \geq 1 \}. \] The family \(\mathcal{U}\) of all sets \(U((C_n), (r_n))\) for all \((C_n) \in \mathcal{C}\) and \((r_n) \in \mathcal{R}\) is a neighborhood base for zero of a locally convex topology on \(M_a(S)\). This topology is denoted by \(\beta^1(S)\). Observe \(\beta^1(S) \leq n(S)\). Denote by \(\kappa(S)\) the locally convex topology on \(M_a(S)\) generated by the seminorms \(P_C(\mu) = | \mu | (C)\), where \(C\) runs through all compact subsets of \(S\). The author shows that the topology \(\beta^1(S)\) on \(M_a(S)\) is the mixed topology with respect to \(\kappa(S)\) and \(n(S)\). Using this result, a new proof is given to show that \((M_a(S), \beta^1(S))\) is a bornological, barrelled or a metrizable space precisely when \(S\) is compact, cf. [the author and \textit{R. Nasr-Isfahani}, Acta Math. Sin., Engl. Ser. 27, No. 5, 933--942 (2011; Zbl 1222.43003)]. A locally compact semigroup \(S\) is called foundation if the closure of the set \(\bigcup\{ \operatorname{supp}(\mu) : \mu \in M_a(S) \}\) coincides with \(S\). The semigroup \(S\) is called compactly cancellative if \(C^{-1}D\) and \(CD^{-1}\) are compa…
reviewed by: Michael J. Puls
Mathematics Subject Classification ID: 22A20
zbMATH DE Number: 6147877
zbMATH Keywords: locally compact semigroup
zbMATH Keywords: mixed topology
zbMATH Keywords: locally convex topology
zbMATH Keywords: semigroup algebra
zbMATH Keywords: semi-topological algebra
author: S. Maghsoudi
MaRDI profile type: Publication
full work available at URL: s00233-012-9395-1
OpenAlex ID: W1986963174
cites work: Algebras of Measures on a Locally Compact Semigroup Iii
cites work: Q3755956
cites work: Q3876158
cites work: Q4727659
cites work: Q3704373
cites work: Q4824686
cites work: Topological modules of continuous homomorphisms
cites work: Q5447324
cites work: Q3376375
cites work: Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup
cites work: Arens regularity of semigroup algebras with certain locally convex topologies
cites work: Arens multiplication on Banach algebras related to locally compact semigroups
cites work: The dual of semigroup algebras with certain locally convex topologies
cites work: Linear spaces with mixed topology
DOI: 10.1007/S00233-012-9395-1
Recommended article: Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup
Recommended article: A Characterisation of Absolutely Continuous Measures on Topological Semigroups
Recommended article: Existence of measures invariant with respect to a semigroup of continuous open mappings of a locally compact topological space
Recommended article: Q4954565
Recommended article: Invariant measures on topological semigroups which have an ideal with open translation mappings
Recommended article: Q3722743
Recommended article: Mixed \(\theta\)-continuity on generalized topological spaces
Recommended article: Q3819689
Recommended article: Topological invariant means on locally compact semigroups
Recommended article: Q4034919
description: scientific article; zbMATH DE number 6147877