On asymmetry of topological centers of the second duals of Banach algebras. (English) Zbl 0894.46034

Summary: Let \({\mathfrak A}\) be a Banach algebra with a bounded approximate identity and let \(Z_{1}(\mathfrak{A}^{**})\) and \(Z_{2}(\mathfrak{A}^{**})\) be the left and right topological centers of \(\mathfrak{A}^{**}\). It is shown that i) \(\mathfrak{A}^{*}\mathfrak{A} = \mathfrak{A} \mathfrak{A}^{*}\) is not sufficient for \(Z_1({\mathfrak A}^{**}) = Z_{2}({\mathfrak A}^{**})\); ii) the inclusion \(\widehat{\mathfrak A } Z_{1}(\mathfrak A ^{**}) \subseteq \widehat{\mathfrak A }\) is not sufficient for \(Z_2({\mathfrak A}^{**}) \widehat{\mathfrak A} \subseteq \widehat{\mathfrak A}\); iii) \(Z_1({\mathfrak A}^{**}) = Z_2({\mathfrak A}^{**}) = \widehat{\mathfrak{A}}\) is not sufficient for \(\mathfrak{A}\) to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali Ülger.


46H20 Structure, classification of topological algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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