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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On asymmetry of topological centers
of the second duals of Banach algebras


Authors: F. Ghahramani, J. P. McClure and M. Meng
Journal: Proc. Amer. Math. Soc. 126 (1998), 1765-1768
MSC (1991): Primary 46H99
DOI: https://doi.org/10.1090/S0002-9939-98-04286-5
MathSciNet review: 1443387
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Abstract: 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 $\hat {\mathfrak{A}} Z_{1}(\mathfrak{A}^{**}) \subseteq \hat {\mathfrak{A}}$ is not sufficient for $Z_{2}(\mathfrak{A}^{**}) \hat {\mathfrak{A}} \subseteq \hat {\mathfrak{A}}$; iii) $Z_{1}(\mathfrak{A}^{**}) = Z_{2}(\mathfrak{A}^{**}) = \hat {\mathfrak{A}}$ is not sufficient for $\mathfrak{A}$ to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali Ülger.


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Additional Information

F. Ghahramani
Affiliation: Department of Mathematics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada

J. P. McClure
Affiliation: Department of Mathematics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada

M. Meng
Affiliation: Department of Mathematics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada

DOI: https://doi.org/10.1090/S0002-9939-98-04286-5
Keywords: Arens product, topological center, Banach module
Received by editor(s): December 5, 1996
Additional Notes: The first author was supported by NSERC grant OGP 003664 and the second author by NSERC grant A8069.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society