Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. R. Arens, `The adjoint of a bilinear operation', Proc. Amer. Math. Soc. 2 (1951), 839 - 848. MR 13:659f
  • 2. H.G. Dales, `The uniqueness of the functional calculus', Proc. London Math. Soc. (3) 27 (1973), 638-648. MR 48:12062
  • 3. J. Duncan and S.A.R. Hosseiniun, `The second dual of a Banach algebra', Proc. Roy. Soc. Edinburgh Sect. A84 (1979), 309 - 325. MR 81f:46057
  • 4. A.T. Lau and V. Losert, On the second conjugate algebra of $L^{1}(G)$ of a locally compact group, J. London Math Soc. 37 (1988), 464-470. MR 89e:43007
  • 5. A.T. Lau and A. Ülger, `Topological centers of certain dual algebras', Trans. Amer. Math. Soc. vol. 348, no. 3 (1996), 1191 - 1212. MR 96h:43003

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46H99

Retrieve articles in all journals with MSC (1991): 46H99


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

American Mathematical Society