Module approximate biprojectivity and module approximate biflatness of Banach algebras
type: Item
instance of: scholarly article
title: Module approximate biprojectivity and module approximate biflatness of Banach algebras
zbMATH Open document ID: 1469.46073
author: Abasalt Bodaghi
author: Somaye Grailoo Tanha
publication date: 2021-05-04T00:00:00Z
review text: The connection between amenability of a locally compact group \(G\) and amenability of the group algebra \(L^1(G)\) was studied by \textit{B. E. Johnson} [Cohomology in Banach algebras. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0256.18014)]. Johnson's theorem is not valid for semigroups, in general. A generalization of Johnson's amenability theorem that closes this gap for semigroups is the notion of module amenability [\textit{M. Amini}, Semigroup Forum 69, No. 2, 243--254 (2004; Zbl 1059.43001)]. Results similar to Johnson's theorem have been obtained for pseudo-amenability and pseudo-contractibility of the group algebra \(L^1(G)\) [\textit{F. Ghahramani} and \textit{Y. Zhang}, Math. Proc. Camb. Philos. Soc. 142, No. 1, 111--123 (2007; Zbl 1118.46046)] and module versions of these notions for the (inverse) semigroup algebra \(L^1(S)\) [\textit{A. Bodaghi} and \textit{A. Jabbari}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 63, No. 3, 449--461 (2017; Zbl 1438.46058)]. The link between module biprojectivity or module biflatness of the semigroup algebra \(L^1(S)\) and the inverse semigroup \(S\) has been studied in [\textit{A. Bodaghi} and \textit{M. Amini}, Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 75, No. 3, 25--36 (2014; Zbl 1299.43002)]. For more on biprojectivity or biflatness of Banach algebras and approximate versions of these notions, see [\textit{A. Ya. Helemskii}, in: Handbook of algebra. Volume 2. Amsterdam: North-Holland. 151--274 (2000; Zbl 0968.46061)] and [\textit{Y. Zhang}, Proc. Am. Math. Soc. 127, No. 11, 3237--3242 (1999; Zbl 0931.46035); \textit{E. Samei} et al., Can. J. Math. 62, No. 4, 845--869 (2010; Zbl 1201.43002); \textit{O. Yu. Aristov}, Sb. Math. 196, No. 11, 1553--1583 (2005; Zbl 1153.46030); translation from Mat. Sb. 196, No. 11, 3--32 (2005); \textit{H. Pourmahmood-Aghababa}, Bull. Aust. Math. Soc. 87, No. 1, 158--173 (2013; Zbl 1276.46039)], respectively. Johnson's theorem has been exten…
reviewed by: Sedigheh Barootkoob
Mathematics Subject Classification ID: 46M10
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 7343273
zbMATH Keywords: inverse semigroup
zbMATH Keywords: module amenability
zbMATH Keywords: module approximate biprojectivity
zbMATH Keywords: module approximate biflatness
MaRDI profile type: Publication
full work available at URL: s12215-020-00503-8
OpenAlex ID: W3088510070
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DOI: 10.1007/S12215-020-00503-8
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 7343273