Derivations on the module extension Banach algebras

  • A. Teymouri Dep. Math., Central Tehran Branch, Islamic Azad Univ., Tehran, Iran
  • A. Bodaghi Dep. Math., Garmsar Branch, Islamic Azad Univ., Garmsar, Iran
  • D. Ebrahimi Bagha  Dep. Math., Central Tehran Branch, Islamic Azad Univ., Tehran, Iran
Keywords: Amenability, Banach algebra, Derivation, Module extension

Abstract

UDC 517.986

We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali,  Ideal amenability of module extension Banach algebras, Int. J. Contemp. Math. Sci.,  2, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension $\mathcal A\oplus X$ to be $(\mathcal I\oplus Y)$-weakly amenable, where $\mathcal I$ is a closed ideal of the Banach algebra $\mathcal A$ and $Y$ is a closed $\mathcal A$-submodule of the Banach $\mathcal A$-bimodule $X.$
We apply this result to the module extension $\mathcal A\oplus(X_1\dotplus X_2),$ where $X_1,$ $X_2$ are two Banach $\mathcal A$-bimodules.

References

W. G. Bade, P. G. Curtis, H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc., 55, 359 – 377 (1987), https://doi.org/10.1093/plms/s3-55_2.359 DOI: https://doi.org/10.1093/plms/s3-55_2.359

H. G. Dales, F. Ghahramani, N. Grønbæk, Derivations into iterated duals of Banach algebras, Stud. Math., 128, 19 – 54 (1998).

H. G. Dales, A. T.-M. Lau, D. Strauss, Banach algebras on semigroups and their compactifications, Mem. Amer. Math. Soc., 205 (2010), https://doi.org/10.1090/S0065-9266-10-00595-8 DOI: https://doi.org/10.1090/S0065-9266-10-00595-8

M. Eshaghi Gordji, F. Habibian, A. Rejali, Ideal amenability of module extension of Banach algebras, Arch. Math. (Brno), 43, 177 – 184 (2007).

M. Eshaghi Gordji, F. Habibian, A. Rejali, Ideal amenability of module extension Banach algebras, Int. J. Contemp. Math. Sci., 2, № 5, 213 – 219 (2007), https://doi.org/10.12988/ijcms.2007.07014 DOI: https://doi.org/10.12988/ijcms.2007.07014

M. E. Gordji, B. Hayati, S. A. R. Hosseinioun, Ideal amenability of Banach algebras and some hereditary properties, J. Sci. Islam. Repub. Iran, 21, № 14, 333 – 341 (2010).

M. E. Gorgi, T. Yazdanpanah, Derivations into duals of Banach algebras, Proc. Indian Acad. Sci. Math. Sci., 114, № 4, 399 – 408 (2004), https://doi.org/10.1007/BF02829444 DOI: https://doi.org/10.1007/BF02829444

N. Grønbæk, B. E. Johnson, G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math., 87, 289 – 324 (1994), https://doi.org/10.1007/BF02773000 DOI: https://doi.org/10.1007/BF02773000

P. R. Halmos, Commutators of operators, II, Amer. J. Math., 76, 191 – 198 (1954), https://doi.org/10.2307/2372409 DOI: https://doi.org/10.2307/2372409

B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972). DOI: https://doi.org/10.1090/memo/0127

B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc., 23, 281 – 284 (1991), https://doi.org/10.1112/blms/23.3.281 DOI: https://doi.org/10.1112/blms/23.3.281

A. Minapoor, A. Bodaghi, D. Ebrahimi Bagha, Ideal Connes-amenability of dual Banach algebras, Mediterr. J. Math., 14, № 174 (2017), https://doi.org/10.1007/s00009-017-0970-2 DOI: https://doi.org/10.1007/s00009-017-0970-2

A. Minapoor, A. Bodaghi, D. Ebrahimi Bagha, Derivations on the tensor product of Banach algebras, J. Math. Ext., 10, № 4, 117 – 125 (2017).

C. Pearcy, D. Topping, On commutators in ideals of compact operators, Michigan Math. J., 18, 247 – 252 (1971). DOI: https://doi.org/10.1307/mmj/1029000686

A. Teymouri, A. Bodaghi, D. Ebrahimi Bagha, Derivations into annihilators of the ideals of Banach algebras, Demonstr. Math., 52, 20 – 28 (2019), https://doi.org/10.1515/dema-2019-0004 DOI: https://doi.org/10.1515/dema-2019-0004

S. Wassermann, On tensor products of certain group $Cast$ -algebras, J. Funct. Anal., 23, 28 – 36 (1976), https://doi.org/10.1016/0022-1236(76)90050-1 DOI: https://doi.org/10.1016/0022-1236(76)90050-1

Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc., 354, № 10, 4131 – 4151 (2002), https://doi.org/10.1090/S0002-9947-02-03039-8 DOI: https://doi.org/10.1090/S0002-9947-02-03039-8

Published
21.04.2021
How to Cite
Teymouri , A., A. Bodaghi, and D. Ebrahimi Bagha. “Derivations on the Module Extension Banach Algebras”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 4, Apr. 2021, pp. 566 -76, doi:10.37863/umzh.v73i4.240.
Section
Research articles