Arens regularity of some bilinear maps
DOI:
https://doi.org/10.4067/S0716-09172009000100002Keywords:
Banach algebra, Bilinear map, Arens products.Abstract
Let H be a Hilbert space. we show that the following statements are equivalent: (a) B(H) is finite dimension, (b) every left Banach module action l : B(H)×H → H, is Arens regular (c) every bilinear map f : B(H)*→ B(H) is Arens regular. Indeed we show that a Banach space X is reflexive if and only if every bilinear map f : X* × X → X* is Arens regular.References
[A] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2(1951), pp. 839-848, (1951).
[D-R-V] H. G. Dales, A. Rodriguez-Palacios and M. V. Velasco, The second transpose of a derivation, J. London Math. Soc. (2) 64, pp. 707-721, (2001).
[D] M. Daws, Arens regularity of the algebra of operators on a Banach space. Bull. London Math. Soc. 36, No. 4, pp. 493—503, (2004).
[D-H] J. Duncan and S. A. Hosseiniun, The second dual of Banach algebra, Proc. Roy. Soc. Edinburgh Set. A 84, pp. 309-325, (1979).
[E-F] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math., Vol 181, No 3, pp. 237-254, (2007).
[F-S] M. Filali and A. I. Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras. General topological algebras (Tartu, 1999), pp. 95— 124, Math. Stud. (Tartu), 1, Est. Math. Soc., Tartu, 2001. MR1853838
[M-Y] Gh. Mohajeri-Minaei, T. Yazdanpanah, Arens regularity of modules. J. Inst. Math. Comput. Sci. Math. Ser. 18, No. 3, pp. 195—197. MR2194165, (2005).
[Pa] Theodore W. Palmer, Banach Algebra and The General Theory of *-Algebras Vol 1 Cambridge University Press,(1994).
[D-R-V] H. G. Dales, A. Rodriguez-Palacios and M. V. Velasco, The second transpose of a derivation, J. London Math. Soc. (2) 64, pp. 707-721, (2001).
[D] M. Daws, Arens regularity of the algebra of operators on a Banach space. Bull. London Math. Soc. 36, No. 4, pp. 493—503, (2004).
[D-H] J. Duncan and S. A. Hosseiniun, The second dual of Banach algebra, Proc. Roy. Soc. Edinburgh Set. A 84, pp. 309-325, (1979).
[E-F] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math., Vol 181, No 3, pp. 237-254, (2007).
[F-S] M. Filali and A. I. Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras. General topological algebras (Tartu, 1999), pp. 95— 124, Math. Stud. (Tartu), 1, Est. Math. Soc., Tartu, 2001. MR1853838
[M-Y] Gh. Mohajeri-Minaei, T. Yazdanpanah, Arens regularity of modules. J. Inst. Math. Comput. Sci. Math. Ser. 18, No. 3, pp. 195—197. MR2194165, (2005).
[Pa] Theodore W. Palmer, Banach Algebra and The General Theory of *-Algebras Vol 1 Cambridge University Press,(1994).
How to Cite
[1]
M. E. Gordji, “Arens regularity of some bilinear maps”, Proyecciones (Antofagasta, On line), vol. 28, no. 1, pp. 21-26, 1.
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