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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.

MSC:

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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[1] R. Arens, ‘The adjoint of a bilinear operation’, Proc. Amer. Math. Soc. 2 (1951), 839 - 848. · Zbl 0044.32601
[2] H. G. Dales, The uniqueness of the functional calculus, Proc. London Math. Soc. (3) 27 (1973), 638 – 648. · Zbl 0269.46038
[3] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309 – 325. · Zbl 0427.46028
[4] Anthony To Ming Lau and Viktor Losert, On the second conjugate algebra of \?\(_{1}\)(\?) of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464 – 470. · Zbl 0608.43002
[5] Anthony To Ming Lau and Ali Ülger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1191 – 1212. · Zbl 0859.43001
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