Abstract
Let \(\mathcal{{A}}\) be a Banach algebra and let \(\mathcal{{X}}\) be an introverted closed subspace of \(\mathcal{{A}}^*\). Here, we give necessary and sufficient conditions for that the dual algebra \(\mathcal{{X}}^*\) of \(\mathcal{{X}}\) or the topological centers \({\mathfrak {Z}}_t^{(1)}(\mathcal{{X}}^{*})\) and \({\mathfrak {Z}}_t^{(2)}(\mathcal{{X}}^{*})\) of \(\mathcal{{X}}^*\) are Banach \(*\)-algebras. We finally apply these results to the Banach space \(L_0^\infty (G)\) of all equivalence classes of essentially bounded functions vanishing at infinity on a locally compact group \(G\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Abtahi, R. Nasr-Isfahani, A. Rejali, Weighted \(L^p\)-conjecture for locally compact groups. Period. Math. Hung. 60, 1–11 (2010)
F.F. Bonsall, J. Duncan, Complete Normed Algebras (Springer-Verlag, New York, 1973)
P. Civin, B. Yood, The second conjugate space of a Banach algebra as an algebra. Pac. J. Math. 11, 847–870 (1961)
H.G. Dales, Banach Algebras and Automatic Continuity (Oxford University Press, New York, 2000)
H.G. Dales, A.T. Lau, The Second Duals of Beurling Algebras (Memoirs of the American Mathematical Society, Providence, 2005)
J. Duncan, S.A.R. Hosseiniun, The second dual of a Banach algebra. Proc. R. Soc. Edinb. 84, 309–325 (1979)
H. Farhadi, F. Ghahramani, Involutions on the second duals of group algebras and a multiplier problem. Proc. Edinb. Math. Soc. 50, 153–161 (2007)
M. Filali, M.S. Monfared, Finite dimensional left ideals in the dual of introverted spaces. Proc. Am. Math. Soc. 139, 3645–3656 (2011)
M. Filali, A.I. Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras. Gen. Topol. Algebra, Math. Stud 1, 95–124 (2001)
M. Grosser, Algebra involutions on the bidual of a Banach algebra. Manuscripta Math. 48, 291–295 (1984)
A.T. Lau, J. Pym, Concerning the second dual of the group algebra of alocally compact group. J. Lond. Math. Soc. 41, 445–460 (1990)
A.T. Lau, A. Ülger, Topological centers of certain dual algebras. Trans. Am. Math. Soc. 348, 1191–1212 (1996)
M.J. Mehdipour, R. Nasr-Isfahani, Compact left multipliers on Banach algebras related to locally compact groups. Bull. Aust. Math. Soc. 79, 227–238 (2009)
M.J. Mehdipour, R. Nasr-Isfahani, Weakly compact multipliers on Banach algebras related to a locally compact group. Acta Math. Hung. 127, 195–206 (2010)
M. Neufang, Solution to Farhadi–Ghahramani’s multiplier problem. Proc. Am. Math. Soc. 138, 553–555 (2010)
A.I. Singh, \(L_0^\infty (G)^\ast \) as the second dual of the group algebra \(L^1(G)\) with a locally convex topology. Mich. Math. J. 46, 143–150 (1999)
A.I. Singh, Involutions on the second duals of group algebras versus subamenable groups. Stud. Math. 206, 51–62 (2011)
Acknowledgments
The authors would like to thank the referee of this paper for valuable remarks. They acknowledge that this research was partially carried out at the IPM-Isfahan Branch. This research was in part supported by a grant from IPM (No. 91430417).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Akhtari, F., Nasr-Isfahani, R. Involutions on certain Banach algebras related to locally compact groups. Period Math Hung 68, 143–149 (2014). https://doi.org/10.1007/s10998-014-0022-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-014-0022-7