Abstract
The concept of left character Connes-amenability for a dual Banach algebra \({\mathcal {A}}\) is introduced. We obtain a cohomological characterization of left character Connes-amenability as well as the relation between left \(\varphi \)-Connes-amenability and existence of left \(\varphi \)-normal virtual diagonals for a \(\omega ^{*}\)-continuous character \(\varphi \). We prove that left character amenability of \({\mathcal {A}}\) is equivalent to left character Connes-amenability of \({\mathcal {A}}^{**}\) when \({\mathcal {A}}\) is Arens regular. Moreover for a locally compact group G, we show that M(G) is left character Connes-amenable. In addition by means of some examples we show that for the new notion, the corresponding class of dual Banach algebras is larger than Connes-amenable dual Banach algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Connes, Classification of injective factors. Ann. Math. 104, 73–114 (1976)
A. Connes, On the cohomology of operator algebras. J. Funct. Anal. 28, 248–253 (1978)
H.G. Dales, Banach Algebras and Automatic Continuity (Clarendon Press, Oxford, 2000)
H.G. Dales, F. Ghahramani, A.Y. Helemskii, The amenability of measure algebras. J. Lond. Math. Soc. 66(2), 213–226 (2002)
H.G. Dales, A.T.M. Lau, D. Strauss, Banach algebras on semigroups and their compactifications. Mem. Am. Math. Soc. 205(966), vi+165 (2010)
M. Daws, Connes-amenability of bidual and weighted semigroup algebras. Math. Scand. 99(2), 217–246 (2006)
E. Kaniuth, A.T. Lau, J. Pym, On \(\varphi -\)amenability of Banach algebras. Math. Proc. Camb. Philos. Soc. 144, 85–96 (2008)
E.G. Effros, Amenability and virtual diagonals for von Neumann algebras. J. Funct. Anal. 78, 137–153 (1988)
U. Haagerup, All nuclear \(C^*\)-algebras are amenable. Invent. Math. 74, 305–319 (1983)
A. Ya. Helemskii, The Homology of Banach and Topological Algebras (translated from the Russian) (Kluwer Academic Publishers, Dordrecht, 1989)
A. Ya. Helemskii, Homlogical essence of amenability in the sence of A. Connes: the injectivity of the predual bimodule (translated from the Russion), Math. USSR-Sb 68, 555–566 (1991)
B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc.127, 1–96 (1972)
B.E. Johnson, R.V. Kadison, J. Ringrose, Cohomology of operator algebras, III. Bull. Soc. Math. France 100, 73–79 (1972)
V. Runde, Amenability for dual Banach algebras. Stud. Math. 148, 47–66 (2001)
V. Runde, Lectures on Amenability, vol. 1774, Lecture Notes in Mathematics (Springer-Verlage, Berlin, 2002)
V. Runde, Connes-amenability and normal virtual diagonals for measure alebras I. J. Lond. Math. Soc. 67, 643–656 (2003)
V. Runde, Connes-amenability and normal virtual diagonals for measure algebras II. Bull. Aust. Math. Soc. 68, 325–328 (2003)
V. Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonal, and injectivity of the predual bimodule. Math. Scand. 95, 124–144 (2004)
V. Runde, A Connes-amenable dual Banach algebra need not hane a normal, virtual diagonal. Trans. Am. Math. Soc. 358, 391–402 (2006)
M. Sangani-Monfared, Character amenability of Banach algebras. Math. Proc. Camb. Philos. Soc. 144, 697–706 (2008)
M. Sangani-Monfared, On certain products of Banach algebras with applications to harmonic analysis. Stud. Math. 178, 227–294 (2007)
B. Shojaee, G.H. Esslamzadeh, A. Pourabbas, First order cohomology of \(\ell ^{1}\)-Munn algebras. Bull. Iran. Math. Soc. 35, 211–219 (2009)
Acknowledgements
The authors would like to thank the anonymous for the comments to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shojaee, B., Amini, M. Character Connes-amenability of dual Banach algebras. Period Math Hung 74, 31–39 (2017). https://doi.org/10.1007/s10998-016-0176-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-016-0176-6