Abstract
The main purpose of this paper is to obtain some results on the image of \(\sigma \)-derivations on Banach algebras. One of the main results of this paper is to prove that if \(\mathcal{A}\) is a commutative Banach algebra and \(d:\mathcal{A} \to \mathcal{A}\) is a continuous \(\sigma \)-derivation such that \(\sigma \) is a continuous homomorphism, \(d\sigma = \sigma d = d\) and \({{\sigma }^{2}} = \sigma \), then \(d(\mathcal{A}) \subseteq \operatorname{rad} (\mathcal{A})\), where \(\operatorname{rad} (\mathcal{A})\) denotes the Jacobson radical of \(\mathcal{A}\). Moreover, we obtain Sinclair’s theorem for \(\sigma \)-derivations without assuming continuity. Indeed, under certain conditions, we prove that if \(d\) is a \(\sigma \)-derivation on a Banach algebra \(\mathcal{A}\), then \(d(\mathcal{P}) \subseteq \mathcal{P}\) for every primitive ideal \(\mathcal{P}\) of \(\mathcal{A}\). Some other related results are also discussed.
Similar content being viewed by others
REFERENCES
H. G. Dales, P. Aiena, J. Eschmeier, K. Laursen, and G. A. Willis, Introduction to Banach Algebras, Operators and Harmonic Analysis (Cambridge Univ. Press, Cambridge, 2003).
S. Ali, A. Fosner, M. Fosner, and M. S. Khan, “On generalized Jordan triple (α,β)*-derivations and related mappings,” Mediterr. J. Math. 10 (4), 1657–1668 (2013). https://doi.org/10.1007/s00009-013-0277-x
M. Ashraf and N. Rehman, “On (σ,τ)-derivations in prime rings,” Arch. Math. 38 (4), 259–264 (2002).
M. E. Gordji, “A characterization of (σ,τ)-derivations on von Neumann algebras,” Univ. Politeh. Bucharest Sci. Bull., Ser. A: Appl. Math. Phys. 73 (1), 111–116 (2011).
Ö. Gölbaşı and E. Koç, “Notes on Jordan (σ,τ)*-derivations and Jordan triple (σ,τ)*-derivations,” Aequat. Math. 85 (3), 581–591 (2013). https://doi.org/10.1007/s00010-012-0149-7
A. Hosseini, M. Hassani, A. Niknam, and S. Hejazian, “Some results on σ-derivations,” Ann. Funct. Anal. 2 (2), 75–84 (2011).
A. Hosseini, M. Hassani, and A. Niknam, “Generalized σ-derivation on Banach algebras,” Bull. Iran. Math. Soc. 37 (4), 81–94 (2011).
A. Hosseini, “Characterization of some derivations on von Neumann algebras via left centralizers,” Ann. Univ. Ferrara 64 (1), 99–110 (2018). https://doi.org/10.1007/s11565-017-0290-2
A. Hosseini, “On the image, characterization, and automatic continuity of (σ,τ)-derivations,” Arch. Math. 109 (5), 461–469 (2017). https://doi.org/10.1007/s00013-017-1082-8
A. Hosseini, M. Hassani, and A. Niknam, “On the range of a derivation,” Iran. J. Sci. Technol. Trans. A Sci. 38 (A2), 111–115 (2014).
A. Hosseini, “A characterization of derivations on uniformly mean value Banach algebras,” Turk. J. Math. 40 (5), 1058–1070 (2016). https://doi.org/10.3906/mat-1506-92
A. Hosseini, “A new proof of Singer–Wermer theorem with some results on {g,h}-derivations,” Int. J. Nonlinear Anal. Appl. 11 (1), 453–471 (2020). https://doi.org/10.22075/ijnaa.2019.17189.1915
A. Hosseini, “When is a (ϕ,I)-derivation continuous and where can its image be found?,” Asian-Eur. J. Math., Online Ready (2022). https://doi.org/10.1142/S1793557123500316
Y. S. Jung and K. H. Park, “Noncommutative versions of the Singer–Wermer conjecture with linear left θ-derivations,” Acta Math. Sin. Engl. Ser. 24 (11), 1891–1900 (2008). https://doi.org/10.1007/s10114-008-6244-y
T.-K. Lee and C.-K. Liu, “Spectrally bounded ϕ-derivations on Banach algebras,” Proc. Am. Math. Soc. 133 (5), 1427–1435 (2005). https://doi.org/10.1090/S0002-9939-04-07655-5
M. Mathieu, “Where to find the image of a derivation,” Banach Center Publ. 30 (1), 237–249 (1994).
M. P. Thomas, “The image of a derivation is contained in the radical,” Ann. Math. 128 (3), 435–460 (1988). https://doi.org/10.2307/1971432
M. P. Thomas, “Primitive ideals and derivations on non-commutative Banach algebras,” Pac. J. Math. 159 (1), 139–152 (1993).
I. M. Singer and J. Wermer, “Derivations on commutative normed algebras,” Math. Ann. 129, 260–264 (1955). https://doi.org/10.1007/BF01362370
B. E. Johnson, “Continuity of derivations on commutative Banach algebras,” Am. J. Math. 91 (1), 1–10 (1969). https://doi.org/10.2307/2373262
N. Boudi and S. Ouchrif, “On generalized derivations in Banach algebras,” Studia Math. 194, 81–89 (2009). https://doi.org/10.4064/sm194-1-5
H. G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monographs New Series, Vol. 24 (Oxford Univ. Press, New York, 2000).
M. Mirzavaziri and M. S. Moslehian, “Automatic continuity of σ-derivations on C*-algebras,” Proc. Am. Math. Soc. 134 (11), 3319–3327 (2006). https://doi.org/10.1090/S0002-9939-06-08376-6
ACKNOWLEDGMENTS
The author thanks the referee for carefully reading the article and suggesting valuable comments that have improved the quality of this work.
Funding
Moreover, this research will be supported by a grant from Kashmar Higher Education Institute, grant no. 28/1348/1400/578.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
About this article
Cite this article
Hosseini, A. What Can Be Expected from the Image of σ-Derivations on Banach Algebras?. Russ Math. 66, 38–50 (2022). https://doi.org/10.3103/S1066369X22070040
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X22070040