Toshkent iqtisodiyot va pedagogika instituti Tashkent Institute of Economics and Pedagogy
Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
AUTOMORPHISMS AND 2-LOCAL AUTOMORHISM OF THREE-DIMENSIONAL JORDAN ALGEBRAS
F.N.Arzikulov
V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences F.B.Nabijonova Doctoral student of the Department of mathematical analysis and differential equations of Fergana State University.
Uzbekistan.
In 1997, P. Semrl introduced and investigated so-called 2-local derivations and 2-local automorphisms on operator algebras. He described such maps on the algebra B(H) of all bounded linear operators on an infinite-dimensional separable Hilbert space H. Namely, he proved that every 2-local derivation (automorphism) on B(H) is a derivation (respectively an automorphism).
A similar description of 2-local derivations for the finite-dimensional case appeared later in the work of S. Kim and J. Kim. In the paper of Y. Lin and T. Wong 2-local derivations have been described on matrix algebras over finite-dimensional division rings. Sh. Ayupov and K. Kudaybergenov suggested a new technique and have generalized the above-mentioned results mentioned above for arbitrary Hilbert spaces. Namely, they proved that every 2-local derivation on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H is a derivation. A similar result is obtained for automorphisms by them. Sh. Ayupov, K. Kudaybergenov and F. Arzikulov extended the above results for 2-local derivations and gave a proof of the theorem for arbitrary von Neumann algebras.
M.J. Burgos, F.J. Fernandez Polo, J.J. Garces and A.M. Peralta established that every 2-local *-homomorphism from a von Neumann algebra into a C*-algebra is a linear *-homomorphism. These authors also proved that every 2-local Jordan *-homomorphism from a JBW*-algebra into a JB*-algebra is a Jordan *-homomorphism. Later, Sh. Ayupov and K. Kudaybergenov prove that any 2-local automorphism on an arbitrary AW*-algebra without finite type I direct summands is an automorphism.
Sh. Ayupov and K. Kudaybergenov also proved that every 2-local automorphism on a finite-dimensional semi-simple Lie algebra over an algebraically closed field of characteristic zero is an automorphism and showed that each finite-
May 15, 2024
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Toshkent iqtisodiyot va pedagogika instituti Tashkent Institute of Economics and Pedagogy
Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current
ChaLes.hnnoeon.andPru.en
dimensional nilpotent Lie algebra with dimension >2 admits a 2-local automorphism which is not an automorphism.
The present paper is devoted to automorphisms and 2-local automorphisms on Jordan algebras.
Sh. Ayupov, F. Arzikulov, N. Umrzaqov and O. Nuriddinov proved that every 2-local 1-automorphism (i.e. implemented by single symmetries) on a finite-dimensional semi-simple Jordan algebra over an algebraically closed field of characteristics different from 2 is an automorphism.
In the present paper, we investigate 2-local automorphisms on low-dimensional Jordan algebras. Namely, we describe automorphisms and 2-local automorphisms on three-dimensional T9 Jordan algebra .
Let T9 be a Jordan algebra with the following table multiplication
2 2 2 A 1
e = e n = n n = o en =^ n
en
n2 nn = 0
i.e., there-dimensional spin factor.
Theorem 1.1. A linear operator 9 on T9 is an automorphism if and only if the matrix generating 9 is equal to one of the following matric
1 0 0
a 21 a 22 0
2 " a21 — a21a22 a
22 j
Theorem 1.2. Every local automorphism on T9 is an automorphism. Theorem 1.3. Every 2-local automorphism on T9 is an automorphism.
REFERENCES
1. Larson D. R., Sourour A. R. Local derivations and local automorphisms of B(X). Proc. Sympos. Pure Math., Providence, Rhode Island Part 2, 1990. Vol. 51. P. 187{194. URI: http://hdl.handle.net/1828/2373
2. R.Kadison. Local derivations, J.Algebra, 130(1990), 494-509. DOI: https://doi.org/10.1016/0021-8693(90)90095-6.
3. I.Kashuba, M.E.Martin, Deformations of Jordan algebras of dimension four, Journal of Algebra 399, 288-289 (2014).
4. B,Johnson, Local derivations on C*- algebras are derivations, Trans. Amer. Math, Soc., 353 (2001), 313-325. DOI: 10.2307/221975
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