Abstract
In this paper we define the module topological center of the second dual \(\mathcal{A}^{**}\) of a Banach algebra \(\mathcal{A}\) which is a Banach \(\mathfrak{A}\)-module with compatible actions on another Banach algebra \(\mathfrak{A}\). We calculate the module topological center of ℓ 1(S)**, as an ℓ 1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ℓ 1(S)** is ℓ 1(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amini, M.: Module amenability for semigroup algebras. Semigroup Forum 69, 243–254 (2004)
Amini, M.: Corrigendum, Module amenability for semigroup algebras. Semigroup Forum 72, 493 (2006)
Amini, M., Ebrahimi Bagha, D.: Weak module amenability for semigroup algebras. Semigroup Forum 71, 18–26 (2005)
Bonsall, F.F., Duncan, J.: Complete Normed Algebra. Springer, Berlin (1973)
Dales, H.G.: Banach Algebra and Automatic Continuity. Oxford University Press, Oxford (2000)
Duncan, J., Namioka, I.: Amenability of inverse semigroups and their semigroup algebras. Proc. R. Soc. Edinb. A 80, 309–321 (1988)
Filali, M., Salmi, P.: Topological centres of weighted convolution algebras. Bull. Can. Math. Soc. doi:10.4153/CMB-2010-016-x
Ghahramani, F., Loy, R.J., Willis, G.A.: Amenability and weak amenability of second conjugate Banach algebras. Proc. Am. Math. Soc. 124, 1489–1497 (1996)
Gourdeau, F.: Amenablility and the second dual of a Banach algebras. Stud. Math. 125(1), 75–80 (1997)
Lashkaizaheh Bami, M.: The topological centers of LUC(S)* and M a (S)** of certain foundation semigroup. Glasg. Math. J. 42, 335–343 (2000)
Lau, A.T.-M.: Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups. Math. Proc. Camb. Philos. Soc. 99, 273–283 (1986)
Lau, A.T.-M., Losert, V.: On the second conjugate algebra of L 1(G) of a locally compact group. J. Lond. Math. Soc. 37, 464–470 (1988)
Lau, A.T., Ulger, A.: Topological centers of certain dual algebras. Trans. Am. Math. Soc. (3) 348, 1191–1212 (1996)
Rezavand, R., Amini, M., Sattari, M.H., Ebrahimi Bagha, D.: Module Arens regularity for semigroup algebras. Semigroup Forum 77, 300–305 (2008)
Rezavand, R., Amini, M., Sattari, M.H., Ebrahimi Bagha, D.: Corrigendum, Module Arens regularity for semigroup algebras. Semigroup Forum 79, 214–215 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jerome A. Goldstein.
Rights and permissions
About this article
Cite this article
Amini, M., Bodaghi, A. & Ebrahimi Bagha, D. Module amenability of the second dual and module topological center of semigroup algebras. Semigroup Forum 80, 302–312 (2010). https://doi.org/10.1007/s00233-010-9211-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-010-9211-8