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Journal of Lie Theory 15 (2005), No. 2, 393--414
Copyright Heldermann Verlag 2005

Homomorphisms between Lie JC*-Algebras and Cauchy-Rassias Stability of Lie JC*-Algebra Derivations

Chun-Gil Park
Dept. of Mathematics, Chungnam National University, Daejeon 305-764, South Korea


It is shown that every almost linear mapping $h\colon A\rightarrow B$ of a unital Lie JC$^*$-algebra $A$ to a unital Lie JC$^*$-algebra $B$ is a Lie JC$^*$-algebra homomorphism when $h(2^n u\circ y)=h(2^n u)\circ h(y)$, $h(3^n u\circ y)=h(3^n u)\circ h(y)$ or $h(q^n u\circ y)=h(q^n u)\circ h(y)$ for all $y\in A$, all unitary elements $u\in A$ and $n=0,1,2,\cdots$, and that every almost linear almost multiplicative mapping $h\colon A\rightarrow B$ is a Lie JC$^*$-algebra homomorphism when $h(2x)=2h(x)$, $h(3x)=3h(x)$ or $h(qx)qh(x)$ for all $x\in A$. Here the numbers $2,3,q$ depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. Moreover, we prove the Cauchy--Rassias stability of Lie JC$^*$-algebra homomorphisms in Lie JC$^*$-algebras, and of Lie JC$^*$-algebra derivations in Lie JC$^*$-algebras.

Keywords: Lie JC*-algebra homomorphism, Lie JC*-algebra derivation, stability, linear functional equation.

MSC: 17B40, 39B52, 46L05; 17A36

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