Jordan代数上的可乘Jordan导子

设A是Jordan代数,如果映射d:A→A满足任给a,b∈A,都有d(aob)=d(a)o b+aod(b),则称d为可乘Jordan导子.如果A含有一个非平凡幂等p,且A对于p的Peirce分解A=A_1⊕A_(1/2)⊕A_0满足:(1)设ai∈Ai(i=1,0),如果任给t_(1/2)∈A_(1/2),都有a_i○t_(1/2)=0,则a_i=0,则A上的可乘Jordan导子d.如果满足d(p)=0,则d是可加的.由此得到结合代数和三角代数满足一定条件时,其上的任意可乘Jordan导子是可加的.

Multiplicative Jordan Derivations on Jordan Algebras

Let A be a Jordan algebra. If the map d:A→A satisfies d(a o b)=d(a) o b+a o d(b) for all a,b∈ A, then d is called a multiplicative Jordan derivation on A. Our main objective in this note is to prove the following. Suppose A has an idempotent p (p≠0,p≠1) which satisfies that the Peirce decomposition of A with respect to p, A=A₁⊕ A_1/2⊕ A₀, satisfies that
(1) Let a_i∈A_i (i=1,0). If a_i o t_1/2=0 for all t_1/2∈ A_1/2, then a_i=0. If d is any multiplicative Jordan derivation of A which satisfies that d(p)=0, then d is additive. As its application, we get the result that every multiplicative Jordan derivation on some associative algebras and triangular algerbas is additive.