Description of Local Derivations on Jordan Algebras of Dimension Five
Аннотация
In the present paper we investigate local derivations on finite dimensional Jordan algebras. The Gleason--Kahane--\.{Z}elazko 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.