Module approximate biprojectivity and module approximate biflatness of Banach algebras
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 extended to the Beurling (weighted) algebra \(L^1(G,\omega)\) of a locally compact group \(G\) in [\textit{N. Grønbæk}, Trans. Am. Math. Soc. 319, No. 2, 765--775 (1990; Zbl 0701.46035)]. Also, see [Johnson, loc. cit.]. For more on Beurling algebras, see [\textit{H. G. Dales} and \textit{A. T. M. Lau}, The second duals of Beurling algebras. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1075.43003)]. Also, the module version of Johnson's theorem has been investigated for weighted semigroup algebras in [\textit{G. Asgari} et al., Commun. Korean Math. Soc. 34, No. 3, 743--755 (2019; Zbl 1460.46035)]. The relation between the locally compact group \(G\) and pseudo-amenability and approximate biprojectivity of \(L^1(G,\omega)\) has been investigated in [\textit{K. Oustad} and \textit{A. Mahmoodi}, ``On pseudo amenability of Beurling algebras'', Preprint (2017), \url{arXiv:1706.07082}] and [\textit{A. Sahami} and \textit{A. Pourabbas}, Colloq. Math. 145, No. 2, 273--284 (2016; Zbl 1352.43003)], respectively. In this paper, motivated by the definitions of module biflatness, module biprojectivity, approximate biflatness, and approximate biprojectivity, the concepts of module approximate biflatness, and module approximate biprojectivity are introduced and it is shown that for an inverse semigroup \(S\) with the set of idempotents \(E\), the concepts of module approximate biprojectivity and module pseudo-amenability for \(L^1(S,\omega)\) as an \(L^1(E)\)-module coincide, and module approximate biflatness and module approximate biprojectivity of it are related to amenability of \(S\). Indeed, necessary and sufficient conditions for module approximate biprojectivity and module approximate bifatness of the Beurling semigroup algebra \(L^1(S,\omega)\) are obtained. Finally, the module approximate biflatness of the projective tensor product of Banach algebras \(A\) and \(B\) is investigated from the point of view of module approximate biflatness of~\(A\).
scientific article; zbMATH DE number 7343273
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