On the topological centre of \(L^{1}(G/H)^{\ast\ast}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1247.43002
DOI: 10.1017/S0004972712000123
published in: Bulletin of the Australian Mathematical Society
publication date: 2012-09-05T00:00:00Z
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 6078020
zbMATH Keywords: homogeneous space
zbMATH Keywords: topological centre
zbMATH Keywords: Arens product
zbMATH Keywords: factorisation property
zbMATH Keywords: Mazur property
author: R. Raisi Tousi
author: R. A. Kamyabi-Gol
author: H. R. Ebrahimi Vishki
MaRDI profile type: Publication
cites work: Solution to a conjecture by Hofmeier-Wittstock
cites work: On a conjecture by Ghahramani-Lau and related problems concerning topological centres
cites work: A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis
cites work: Q4520727
cites work: Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups
cites work: Decomposability of von Neumann Algebras and the Mazur Property of Higher Level
cites work: On the Second Conjugate Algebra of L 1 (G ) of a Locally Compact Group
full work available at URL: s0004972712000123
OpenAlex ID: W2020349436
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title: On the topological centre of \(L^{1}(G/H)^{\ast\ast}\)
review text: Let \(G\) be a locally compact group and \(H\) be a compact subgroup of \(G\). This paper is devoted to show that the Banach algebra \(L^1(G/H)\) is strongly Arens irregular for a large class of locally compact groups.NEWLINENEWLINENEWLINENEWLINEReviewer's remark: The reader(s) should be warned that the paper is full of inaccuracies and statements which are based on mathematically false results. Among the facts which are insufficiently taken into account; (i) Proposition 3.3 of [\textit{R. A. Kamyabi-Gol} and \textit{N. Tavallaei}, Bull. Iran. Math. Soc. 35, No. 1, 129--146 (2009; Zbl 1184.43003)] is simply false for general homogeneous spaces. It can be checked readily that Proposition 3.3 of [loc. cit.] is true if and only if \(H\) is normal in \(G\). (ii) Thus, invoking (i), the convolution (resp. involution) defined in [loc. cit.] are well-defined if and only if \(H\) is a normal subgroup of \(G\) and hence in this case the convolution (resp. involution) coincide with the standard \(\ast\)-algebra structure of the Banach function space \(L^1(G/H)\) via the group structure of \(G/H\). (iii) Concerning the convolution over the Banach measure space \(M(G/H)\), the convolution given by (1.3) is well-defined if and only if \(H\) is a normal subgroup of \(G\), hence in this case the convolution (resp. involution) coincide with the standard \(\ast\)-algebra structure of the Banach measure space \(M(G/H)\) via the group structure of \(G/H\).
reviewed by: Arash Ghaani Farashahi
description: scientific article; zbMATH DE number 6078020