Abstract
Let \(\mathbb{G}\mathbb{L}_{n}(\mathbb {C})\) be the complex general linear group of degree n. In this paper, we present the general form of all continuous endomorphisms of \(\mathbb{G}\mathbb{L}_{2}(\mathbb {C})\) with respect to the Jordan triple product. These are the continuous maps \( \varphi :\mathbb{G}\mathbb{L}_{2}(\mathbb {C}) \rightarrow \mathbb{G}\mathbb{L}_{2}(\mathbb {C})\) which satisfy
As a result, we present the general form of all continuous homomorphisms and automorphisms of \(\mathbb{G}\mathbb{L}_{2}(\mathbb {C})\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
13 June 2023
An Erratum to this paper has been published: https://doi.org/10.1007/s13226-023-00438-7
References
T. Abe, S. Akiyama, O. Hatori, Isometries of the special orthogonal group, Linear Algebra Appl., 439 (2013) 174-188.
J. Aczél, J. Dhombres, Functional equations in several variables, Encyclopedia Math. Appl. 31, Cambridge Univ. Press, Cambridge, 1989.
M. Dobovišek, Maps from \(M_{2}(\mathbb{F})\) to \(M_{3}(\mathbb{F})\) that are multiplicative with respect to the Jordan triple product, Aequat. Math., 85 (2013) 539-552.
W. H. Gustafson, P. R. Halmos, H. Radjavi, Products of involutions, Linear Algebra Appl., 13, (1976) 157-162.
O. Hatori, G. Hirasawa, T. Miura, L. Molnár, Isometries and maps compatible with inverted Jordan triple products on groups, Tokyo J. Math., 35 (2012) 385-410.
O. Hatori, L. Molnár, Isometries of the unitary group, Proc. Amer. Math. Soc., 140 (2012) 2141-2154.
K. Hoffman, R. A. Kunze, Linear Algebra, Prentice-Hall, 1971.
D. Kokol Bukošek, B. Mojškerc, Jordan triple product homomorphisms on Hermitian matrices of dimension two, Publ. Math. Debrecen, 90 (2017) 227-250.
F. Lu, Jordan triple maps, Linear Algebra Appl., 375 (2003) 311-317.
L. Molnár, P. Šemrl, Transformations of the unitary group on a Hilbert space, J. Math. Anal. Appl., 388 (2012) 1205-1217.
L. Molnár, Jordan triple endomorphisms and isometries of unitary groups, Linear Algebra Appl., 439 (2013) 3518-3531.
P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, 2011.
A. Taghavi, S. Salehi, Continuous maps preserving Jordan triple products from \(\mathbb{G}\mathbb{L}_{n}(\mathbb{C})\) into \(\mathbb{C}^{\ast }\), Indag. Math., 28 (2017) 1233-1239.
A. Taghavi, S. Salehi, Continuous maps preserving Jordan triple products from \(\mathbb{U}_{n}\) to \(\mathbb{D}_{m}\), Indag. Math., 30 (2019) 157-164.
A. Taghavi, S. Salehi, Continuous maps preserving Jordan triple products from \(\mathbb{G}\mathbb{L}_{1}\) to \(\mathbb{G}\mathbb{L}_{2}\) and \(\mathbb{G}\mathbb{L}_{3}\), Linear Multilinear Algebra, 69 (2021) 208-223.
J. Väisälä, A proof of the Mazur-Ulam theorem, Amer. Math. Monthly, 110 (2003) 633-635.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Kaushal Verma.
The original online version of this article was revised: In this article some typographical mistakes are corrected.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ghasempouri, S.E., KhaliliAsboei, A. & SalehiAmiri, S.S. Continuous Jordan triple endomorphisms of \(\pmb {\mathbb{G}\mathbb{L}_{2}(\mathbb {C})}\). Indian J Pure Appl Math 55, 691–708 (2024). https://doi.org/10.1007/s13226-023-00395-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-023-00395-1