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ARENS REGULARITY OF MODULE ACTIONS AND THE SECOND ADJOINT OF A DERIVATION

Part of: Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  01 June 2008

S. MOHAMMADZADEH  and
H. R. E. VISHKI
S. MOHAMMADZADEH
Affiliation:
Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 91775-1159, Mashhad, Iran (email: somohammadzad@yahoo.com)
H. R. E. VISHKI*
Affiliation:
Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 91775-1159, Mashhad, Iran Centre of Excellence in Analysis on Algebraic Strutures (CEAAS), Ferdowski University of Mashhad, Mashhad, Iran (email: vishki@ferdowsi.um.ac.ir)
*
✉For correspondence; e-mail: vishki@ferdowsi.um.ac.ir
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Abstract

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In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and then investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.

Keywords

Arens productbounded bilinear mapBanach module actionderivationmodule actionsecond dual

MSC classification

Secondary: 46H20: Structure, classification of topological algebras 46H25: Normed modules and Banach modules, topological modules (if not placed in or )
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 3 , June 2008 , pp. 465 - 476
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Akemann, C. A., ‘The dual space of an operator algebra’, Trans. Amer. Math. Soc. 126 (1967), 286–302.CrossRefGoogle Scholar
[2]Akemann, C. A., Dodds, P. G. and Gamlen, J. L. B., ‘Weak compactness in the dual space of a C *-algebra’, J. Funct. Anal. 10 (1972), 446–450.CrossRefGoogle Scholar
[3]Arens, A., ‘The adjoint of a bilinear operation’, Proc. Amer. Math. Soc. 2 (1951), 839–848.CrossRefGoogle Scholar
[4]Arikan, N., ‘Arens regularity and reflexivity’, Quart. J. Math. Oxford 32(2) (1981), 383–388.CrossRefGoogle Scholar
[5]Arikan, N., ‘A simple condition ensuring the Arens regularity of bilinear mappings’, Proc. Amer. Math. Soc. 84 (1982), 525–532.CrossRefGoogle Scholar
[6]Bunce, J. W. and Paschke, W. L., ‘Derivations on a C *-algebra and its double dual’, J. Funct. Anal. 37 (1980), 235–247.CrossRefGoogle Scholar
[7]Civin, P. and Yood, B., ‘The second conjugate space of a Banach algebra as an algebra’, Pacific J. Math. 11 (1961), 847–870.Google Scholar
[8]Dales, H. G., Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, 24 (Clarendon Press, Oxford, 2000).Google Scholar
[9]Dales, H. G. and Lau, A. T.-M., ‘The second duals of Beurling algebras’, Mem. Amer. Math. Soc. 177(836) (2005).Google Scholar
[10]Dales, H. G., Rodríguez-Palacios, A. and Velasco, M. V., ‘The second transpose of a derivation’, J. London Math. Soc. 64(2) (2001), 707–721.CrossRefGoogle Scholar
[11]Ulger, A., ‘Weakly compact bilinear forms and Arens regularity’, Proc. Amer. Math. Soc. 101 (1987), 697–704.CrossRefGoogle Scholar
[12]Young, N. J., ‘The irregularity of multiplication in group algebras’, Quart. J. Math. Oxford 24(2) (1973), 59–62.Google Scholar