Abstract
The main purpose of this paper is to investigate the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras and to present several results in this regard. For instance, we prove the following theorem:
Let \( \mathcal {A}\) and \(\mathcal {B} \) be two Banach algebras such that \(\mathcal {A} \) has the Cohen’s factorization property and \(\bigcap _{\varphi \in \Phi _B} ker (\varphi ) = \{0\} \), and let \(\sigma , \tau :\mathcal {A} \rightarrow \mathcal {B}\) be two linear mappings. Let \( \Delta :\mathcal {A} \rightarrow \mathcal {B} \) be a generalized \((\sigma , \tau ) \)-derivation associated with a \((\sigma , \tau ) \)-derivation \(d:\mathcal {A} \rightarrow \mathcal {B}\). If for any \(\varphi \in \Phi _B \) there exists an element \(a_{\varphi } \in \mathcal {A} \) such that \(\varphi d(a_{\varphi }) \neq 0 \), then \(d \) is continuous if and only if \(\Delta \) is continuous.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
F. F. Bonsall and J. Duncan, Complete Normed Algebras (Springer-Verlag, Berlin-Heidelberg-New York, 1973).
W. G. Bade and P. C. Curtis, “Homomorphisms of commutative Banach algebras,” Amer. J. Math., 82, 589 (1960).
O. Bratteli and D. R. Robinson, Operator Algebras and Quantum Statistical Mechanics. I (Springer Verlag, New York, 1987).
O. Bratteli and D. R. Robinson, Operator Algebras and Quantum Statistical Mechanics. II (Springer Verlag, New York, 1997).
O. Buhler, A Brief Introduction to Classical, Statistical, and Quantum Mechanics (American Mathematical Society, New York, 2006).
H. G. Dales, Banach Algebras and Automatic Continuity (London Math. Soc. Monographs, New Series, 24, Oxford University Press, New York, 2000).
H. G. Dales, P. Aiena, J. Eschmeier, K. Laursen, and G. A. Willis, Introduction to Banach Algebras, Operators and Harmonic Analysis (Cambridge University Press, Cambridge, England, 2003).
O. Elchinger, K. Lundengerd, A. Makhlouf, and S. Silvestrov, “Brackets with \((\tau ,\sigma )\)-derivations and \((p,q) \)-deformations of Witt and Virasoro algebras,” Forum Mathematicum 28, 657 (2016).
M. Gholampour and S. Hejazian, “Continuity of generalized Jordan derivations on semisimple Banach algebras,” Bull. Malays. Math. Sci. Soc. 42, 3273 (2019).
A. Hosseini, M. Hassani, and A. Niknam, “Generalized \(\sigma \)-derivation on Banach algebras,” Bull. Iranian Math. Soc. 37, 81 (2011).
A. Hosseini, “On the image, characterization, and automatic continuity of \((\sigma , \tau )\)-derivations,” Arch. Math. 109, 461 (2017).
J. T. Hartwig, D. Larsson, and S. D. Silvestrov, “Deformations of Lie algebras using \(\sigma \)-derivations,” J. of Algebra 295, 314 (2006).
S. Hejazian, A. R. Janfada, M. Mirzavaziri, and M. S. Moslehian, “Achivement of continuity of \((\phi ,\psi )\) -derivations,” Bull. Belg. Math. Soc. Simon Stevn. 14, 641 (2007).
M. Heller, T. Miller, L. Pysiak, and W. Sasin, “Generalized derivations and general relativity,” Canadian J. of Physics, 91, 757 (2013).
C. Houand Q. Ming, “Continuity of \((\alpha , \beta ) \)-derivations of operator algebras.” J. Korean Math. Soc. 48, 823 (2011).
B. E. Johnson and A. M. Sinclair, “Continuity of derivations and a problem of Kaplansky,” Amer. J. Math. 90, 1067 (1968).
I. Kaplansky, “Functional analysis. Some aspects of analysis and probability,” Surveys in Appl. Math. 4, 1 (1958).
E. Kreyszig, Introductory Functional Analysis with Applications (John Wiley and Sons, New-York, 1978).
M. Mirzavaziri and M. S. Moslehian, “Automatic continuity of \(\sigma \)-derivations on \(C^{\ast } \)-algebras,” Proc. Amer. Math. Soc. 134, 3319 (2006).
M. Mirzavaziri and E. Omidvar Tehrani, “\(\delta \)-double derivations on \(C^{\ast } \)-algebras,” Bull. Iranian. Math. Soc. 35, 147 (2009).
A. M. Peralta and B. Russo, “Automatic continuity of derivations on \(C^{*} \)-algebras and \(JB^{*} \)-triples,” J. Algebra. 399, 960 (2014).
J. R. Ringrose, “Automatic continuity of derivations of operator algebras,” J. London Math. Soc. 5, 432 (1972).
S. Sakai, “On a conjecture of Kaplansky,” Tôhoku Math. J. 12, 31 (1960).
A. R. Villena, “Automatic continuity in associative and nonassociative context,” Irish Math. Soc. Bull. 46, 43 (2001).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Hosseini, A. Contributions to Automatic Continuity of \((\sigma , \tau ) \)-Derivations on Banach Algebras. Sib. Adv. Math. 33, 15–27 (2023). https://doi.org/10.1134/S1055134423010029
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1055134423010029