Abstract
We extend the concept of Arens regularity of a Banach algebra \(\mathcal{A}\) to the case that there is an \(\mathcal{O}\) -module structure on \(\mathcal{A}\) , and show that when S is an inverse semigroup with totally ordered subsemigroup E of idempotents, then A=ℓ 1(S) is module Arens regular if and only if an appropriate group homomorphic image of S is finite. When S is a discrete group, this is just Young’s theorem which asserts that ℓ 1(S) is Arens regular if and only if 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., Ebrahimi Bagha, D.: Weak module amenability for semigroup algebras. Semigroup Forum 71, 18–26 (2005)
Arens, R.: The adjoint of a bilinear operation. Proc. Am. Math. Soc. 2, 839–848 (1951)
Dales, H.G.: Banach Algebras and Automatic Continuity. Clarendon, Oxford (2000)
Dancan, J., Hosseiniun, S.A.R.: The second dual of a Banach algebra. Proc. Edinb. Math. Soc. A 84, 309–325 (1979)
Pym, J.S.: The convolution of function on space of bounded functions. Proc. Lond. Math. Soc. 15, 84–104 (1965)
Young, N.J.: The irregularity of multiplication in group algebras. Q. J. Math. 24, 59–62 (1973)
Young, N.J.: Semigroup algebras having regular multiplication. Stud. Math. 47, 191–196 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jerome A. Goldstein.
An erratum to this article can be found at http://dx.doi.org/10.1007/s00233-009-9157-x
Rights and permissions
About this article
Cite this article
Rezavand, R., Amini, M., Sattari, M.H. et al. Module Arens regularity for semigroup algebras. Semigroup Forum 77, 300–305 (2008). https://doi.org/10.1007/s00233-008-9075-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-008-9075-3