Top

Published in:

2024 | OriginalPaper | Chapter

The Noncommutative Singer-Wermer Conjecture and Generalized Skew Derivations

Authors : Feng Wei, Jing-Xiong Xu

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The noncommutative Singer-Wermer conjecture states that every linear derivation on a noncommutative Banach algebra maps into its Jacobson radical. This conjecture is still an open question for more than thirty years. In this paper, the question of when a generalized skew derivation on a Banach algebra has quasinilpotent values is considered and how this question is related to the noncommutative Singer-Wermer conjecture is discussed.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Local and 2-local Lie-type Derivations of Operator Algebras on Banach Spaces

next chapter Non-global Multiplicative Lie Triple Derivations on Rings

Brešar, M., Villena, A.R.: The noncommutative Singer-Wermer conjecture and \(\phi \)-derivations. J. London Math. Soc. 66, 710–720 (2002)MathSciNetCrossRef

Cusack, J.: Automatic continuity and topologically simple radical Banach algebras. J. London Math. Soc. 16, 493–500 (1977)MathSciNetCrossRef

Dales, H.G.: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, Oxford University Press (2000)

Johnson, B.E.: The uniqueness of the (complete) norm topology. Bull. Amer. Math. Soc. 73, 537–539 (1967)MathSciNetCrossRef

Johnson, B.E.: Continuity of derivations on commutative Banach algebras. Amer. J. Math. 91, 1–10 (1969)MathSciNetCrossRef

Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067–1073 (1968)MathSciNetCrossRef

Jung, Y.-S., Chang, I.-S.: On derivations in Banach algebras. Bull. Korean Math. Soc. 39, 635–643 (2002)MathSciNetCrossRef

Jung, Y.-S., Chang, I.-S.: On approximately higher ring derivations. J. Math. Anal. Appl. 343, 636–643 (2008)MathSciNetCrossRef

Jung, Y.-S., Park, K.-H.: On generalized \((\alpha, \beta )\)-derivations and communitivity in prime rings. Bull. Korean Math. Soc. 43, 101–106 (2006)MathSciNetCrossRef

10.

Jung, Y.-S., Park, K.-H.: Noncommutative versions of the Singer-Wermer conjecture with linear left \(\theta \)-derivations. Acta Math. Sinica 24, 1891–1900 (2008)MathSciNetCrossRef

11.

Kim, B.-D.: Jordan derivations of semiprime rings and noncommutative Banach algebras. II. J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 15, 259–296 (2008)

12.

Kim, B.-D.: Jordan derivations of semiprime rings and noncommutative Banach algebras. I. J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 15, 179–201 (2008)

13.

Kim, B.-D.: On the derivations of semiprime rings and noncommutative Banach algebras. Acta Math. Sinica 16, 21–28 (2000)MathSciNetCrossRef

14.

Mathieu, M.: Where to find the image of a derivation. Banach Center Publ. 30, 237–249 (1994)MathSciNetCrossRef

15.

Park, K.-H., Jung, Y.-S., Bae, J.-H.: Derivations in Banach algebras. Int. J. Math. Math. Sci. 29(10), 579–583 (2002)MathSciNetCrossRef

16.

Sinclair, A.M.: Continuous derivations on Banach algebras. Proc. Amer. Math. Soc. 29, 166–170 (1969)MathSciNetCrossRef

17.

Sinclair, A.M.: Automatic Continuity of Linear Operators, London Mathematical Society Lecture Note Series, vol. 21. Cambridge University Press, Cambridge (1976)CrossRef

18.

Singer, I.M., Wermer, J.: Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955)MathSciNetCrossRef

19.

Thomas, M.P.: The image of a derivation is contained in the radical. Ann. Math. 128, 435–460 (1988)MathSciNetCrossRef

20.

Thomas, M.P.: Primitive ideals and derivations on non-commutative Banach algebras. Pacific J. Math. 159, 139–152 (1993)MathSciNetCrossRef

21.

Vukman, J.: Identities with derivations on rings and Banach algebras. Glas. Mat. Ser. III(40), 189–199 (2005)MathSciNetCrossRef

22.

Wei, F., Xiao, Z.-K.: Pair of derivetions on rings and Banach algebras. Demonstratio Math. 41, 297–308 (2008)MathSciNet

Title: The Noncommutative Singer-Wermer Conjecture and Generalized Skew Derivations
Authors: Feng Wei
Jing-Xiong Xu
Publisher: Springer Nature Switzerland
Book: Advances in Ring Theory and Applications
Print ISBN: 978-3-031-50794-6

Electronic ISBN: 978-3-031-50795-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-50795-3_14

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"