Ultrapowers of Banach algebras
DOI:
https://doi.org/10.21638/spbu01.2022.313Abstract
In this paper, we consider ultrapowers of Banach algebras as Banach algebras and the product (J,U) on the second dual of Banach algebras. For a Banach algebra A, we show that if there is a continuous derivation from A into itself, then there is a continuous derivation from (A**,(J,U)) into it. Moreover, we show that if there is a continuous derivation from A into X**, where X is a Banach A-bimodule, then there is a continuous derivation from A into ultrapower of X i. e., (X)U . Ultra (character) amenability of Banach algebras is investigated and it will be shown that if every continuous derivation from A into (X)U is inner, then A is ultra amenable. Some results related to left (resp. right) multipliers on (A**,(J,U)) are also given.Keywords:
amenability, arens products, derivation, multiplier, ultrapower, ultra amenable, ultra character amenability
Downloads
Download data is not yet available.
References
Литература
1. Heinrich S. Ultraproducts in Banach spaces theory. J. Reine Angew. Math. 313, 72-104 (1980).
2. Godefroy G., Iochum B. Arens-regularity of Banach algebras and geometry of Banach spaces. J. Funct. Anal. 80, 47-59 (1988). https://doi.org/10.1016/0022-1236(88)90064-X
3. Iochum B., Loupias G. Remarks on the bidual of Banach algebras. Colloque Montpellier (1985).
4. Iochum B., Loupias G. Arens regularity and local reflexivity principle for Banach algebras. Math. Ann. 284, 23-40 (1989). https://doi.org/10.1007/BF01443502
5. Daws M. Amenability of ultrapowers of Banach algebras. Proc. Edin. Math. Soc. 52, 307-338 (2009). https://doi.org/10.1017/S0013091507001083
6. Monfared M.S. Character amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144, 697-706 (2008). https://doi.org/10.1017/S0305004108001126
7. Hu Z., Monfared M.S., Traynor T. On character amenable Banach algebras. Studia Math. 193 (1), 53-78 (2009). https://doi.org/10.4064/sm193-1-3
8. Kaniuth E., Lau A.T., Pym J. On φ-amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144, 85-96 (2008). https://doi.org/10.1017/S0305004107000874
9. Alaghmandan M., Nasr-Isfahani R., Nemati M. Character amenability and contractibility of abstract Segal algebras. Bull. Aust. Math. Soc. 82, 274-281 (2010).
10. Dashti M., Nasr-Isfahani R., Renani S.S. Character amenability of Lipschitz algebras. Canad. Math. Bull. 57 (1), 37-41 (2014). https://doi.org/10.4153/CMB-2012-015-3
11. Gordji M.E., Jabbari A., Kim G.H. Some characterization of character amenable Banach algebras. Bull. Korean Math. Soc. 52 (3), 761-769 (2015). https://doi.org/10.4134/BKMS.2015.52.3.761
12. Kaniuth E., Lau A.T., Pym J. On character amenability of Banach algebras. J. Math. Anal. Appl. 344, 942-955 (2008). https://doi.org/10.1016/j.jmaa.2008.03.037
13. Daws M. Ultrapowers of Banach algebras and modules. Glasgow Math. J. 50, 539-559 (2008).
14. Behrends E. A generalization of the principle of local reflexivity. Rev. Roum. Math. Pures Appl. 31, 293-296 (1986).
15. Johnson B.E. Cohomology in Banach algebras. In Ser.: Memoirs of the American Mathematical Society, no. 127. Providence, Rhode Island, Amer. Math. Soc. (1972).
16. Mat´e L. Embedding multiplier operators of a Banach algebra B into its second conjugate space B**. Bull. Acad. Polon. Sei. Ser. Sei. Math. Astronom. Phys. 13, 809-812 (1965).
17. Tomiuk B.J. Multipliers on Banach algebras. Studia Math. 54, 267-283 (1976).
18. Tomiuk B.J. Arens regularity and the algebra of double multipliers. Proc. Amer. Math. Soc. 81 (2), 293-298 (1981).
19. Tomiuk B.J. A correction to “Arens regularity and the algebra of double multipliers”. Proc. Amer. Math. Soc. 91 (1), 171 (1984).
20. Grosser M. Arens semi-regular Banach algebras. Mh. Math. 98, 41-52 (1984).
21. Grosser M., Losert V., Rindler H. Double multipliers’ und asymptotisch invariante approximierende Einheiten. Anz. Osterr. Akad. Wiss. Math.-Naturwiss. ¨ , 7-11 (1980).
22. Kaniuth E. A course in commutative Banach algebras. Springer (2009).
References
1. Heinrich S. Ultraproducts in Banach spaces theory. J. Reine Angew. Math. 313, 72-104 (1980).
2. Godefroy G., Iochum B. Arens-regularity of Banach algebras and geometry of Banach spaces. J. Funct. Anal. 80, 47-59 (1988). https://doi.org/10.1016/0022-1236(88)90064-X
3. Iochum B., Loupias G. Remarks on the bidual of Banach algebras. Colloque Montpellier (1985).
4. Iochum B., Loupias G. Arens regularity and local reflexivity principle for Banach algebras. Math. Ann. 284, 23-40 (1989). https://doi.org/10.1007/BF01443502
5. Daws M. Amenability of ultrapowers of Banach algebras. Proc. Edin. Math. Soc. 52, 307-338 (2009). https://doi.org/10.1017/S0013091507001083
6. Monfared M.S. Character amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144, 697-706 (2008). https://doi.org/10.1017/S0305004108001126
7. Hu Z., Monfared M.S., Traynor T. On character amenable Banach algebras. Studia Math. 193 (1), 53-78 (2009). https://doi.org/10.4064/sm193-1-3
8. Kaniuth E., Lau A.T., Pym J. On φ-amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144, 85-96 (2008). https://doi.org/10.1017/S0305004107000874
9. Alaghmandan M., Nasr-Isfahani R., Nemati M. Character amenability and contractibility of abstract Segal algebras. Bull. Aust. Math. Soc. 82, 274-281 (2010).
10. Dashti M., Nasr-Isfahani R., Renani S.S. Character amenability of Lipschitz algebras. Canad. Math. Bull. 57 (1), 37-41 (2014). https://doi.org/10.4153/CMB-2012-015-3
11. Gordji M.E., Jabbari A., Kim G.H. Some characterization of character amenable Banach algebras. Bull. Korean Math. Soc. 52 (3), 761-769 (2015). https://doi.org/10.4134/BKMS.2015.52.3.761
12. Kaniuth E., Lau A.T., Pym J. On character amenability of Banach algebras. J. Math. Anal. Appl. 344, 942-955 (2008). https://doi.org/10.1016/j.jmaa.2008.03.037
13. Daws M. Ultrapowers of Banach algebras and modules. Glasgow Math. J. 50, 539-559 (2008).
14. Behrends E. A generalization of the principle of local reflexivity. Rev. Roum. Math. Pures Appl. 31, 293-296 (1986).
15. Johnson B.E. Cohomology in Banach algebras. In Ser.: Memoirs of the American Mathematical Society, no. 127. Providence, Rhode Island, Amer. Math. Soc. (1972).
16. Mat´e L. Embedding multiplier operators of a Banach algebra B into its second conjugate space B**. Bull. Acad. Polon. Sei. Ser. Sei. Math. Astronom. Phys. 13, 809-812 (1965).
17. Tomiuk B.J. Multipliers on Banach algebras. Studia Math. 54, 267-283 (1976).
18. Tomiuk B.J. Arens regularity and the algebra of double multipliers. Proc. Amer. Math. Soc. 81 (2), 293-298 (1981).
19. Tomiuk B.J. A correction to “Arens regularity and the algebra of double multipliers”. Proc. Amer. Math. Soc. 91 (1), 171 (1984).
20. Grosser M. Arens semi-regular Banach algebras. Mh. Math. 98, 41-52 (1984).
21. Grosser M., Losert V., Rindler H. Double multipliers’ und asymptotisch invariante approximierende Einheiten. Anz. Osterr. Akad. Wiss. Math.-Naturwiss. ¨ , 7-11 (1980).
22. Kaniuth E. A course in commutative Banach algebras. Springer (2009).
Downloads
Published
2022-10-10
How to Cite
Ebadian, A., & Jabbari, A. (2022). Ultrapowers of Banach algebras. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(3), 527–541. https://doi.org/10.21638/spbu01.2022.313
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
