Description of local derivations on Jordan algebras of dimension five

F. N. Arzikulov   ; O. O. Nuriddinov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 28-43

Résumé

In the present paper we investigate local derivations on finite dimensional Jordan algebras. The Gleason–Kahane–Żelazko theorem, which is a fundamental contribution in the theory of Banach algebras, asserts that every unital linear functional $F$ on a complex unital Banach algebra $A$, such that $F(a)$ belongs to the spectrum $\sigma(a)$ of $a$ for every $a\in A,$ is multiplicative. In modern terminology this is equivalent to the following condition: every unital linear local homomorphism from a unital complex Banach algebra $A$ into ${\Bbb C}$ is multiplicative. We recall that a linear map $T$ from a Banach algebra $A$ into a Banach algebra $B$ is said to be a local homomorphism if, for every $a$ in $A$, there exists a homomorphism $\Phi_a : A\to B$, depending on $a$, such that $T(a)=\Phi_a(a)$. A similar notion was introduced and studied to give a characterization of derivations on operator algebras. Namely, the concept of local derivations was introduced by R. Kadison and D. Larson, A. Sourour independently in 1990. R. Kadison gave the description of all continuous local derivations from a von Neumann algebra into its dual Banach bemodule. B. Jonson extends the result of R. Kadison by proving that every local derivation from a $C^*$-algebra into its Banach bimodule is a derivation. It is known that, every local derivation on a JB-algebra is a derivation. In particular, every local derivation on a finite dimensional semisimple Jordan algebra is a derivation. In the present paper we investigate derivations and local derivations on five-dimensional nilpotent non-associative Jordan algebras. The description of local derivations of nilpotent Jordan algebras is an open problem. In the present paper we give the description of local derivations on five-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of characteristic $\neq 2$, $3$. We also give a criterion of a linear operator on Jordan algebras of dimension five to be a local derivation.

Bibliographie

[1] Gleason A., “A Characterization of Maximal Ideals in Commutative Banach Algebras”, Journal d'Analyse Mathématique, 19 (1967), 171–172 | DOI | MR | Zbl

[2] Kahane J. and Żelazko W., “A Characterization of Maximal Ideals in Commutative Banach Algebras”, Studia Mathematica, 29 (1968), 339–343 | DOI | MR | Zbl

[3] Kadison R., “Local Derivations”, Journal of Algebra, 130 (1990), 494–509 | DOI | MR | Zbl

[4] Larson D. and Sourour A., “Local Derivations and Local Automorphisms of $B(X)$”, Proceedings of Symposia in Pure Mathematics, 51, no. 2, Providence, Rhode Island, 1990, 187–194 | DOI | MR | Zbl

[5] Johnson B., “Local Derivations on $C^*$-Algebras are Derivations”, Transactions of the American Mathematical Society, 353 (2001), 313–325 | DOI | MR | Zbl

[6] Arzikulov F. N. and Umrzaqov N. M., “Local Mappings Generated by a Multiplication on Rings of Matrices”, Mathematical Notes, 107 (2020), 887–897 | DOI | MR | Zbl

[7] Ayupov Sh. A., Kudaybergenov K. K. and Peralta A. M., “A Survey on Local and 2-Local Derivations on $C^*$-and von Neumann Algebras”, Topics in Functional Analysis and Algebra, Contemporary Mathematics, 672, 2016, 73–126 | DOI | MR | Zbl

[8] Hegazi A. S. and Abdelwahab H., “Classification of Five-Dimensional Nilpotent Jordan Algebras”, Linear Algebra and its Applications, 494 (2016), 165–218 | DOI | MR | Zbl

[9] Kashuba I. and Martin M. E., “Geometric Classification of Nilpotent Jordan Algebras of Dimension Five”, Journal of Pure and Applied Algebra, 222:3 (2018), 546–559 | DOI | MR | Zbl

[10] Nuriddinov O., “Description of Local Derivations on Low-Dimension Jordan Algebras”, Bulletin of the Institute of Mathematics, 7:3 (2024), 55–62 | MR