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On homomorphisms between $C^*$-algebras and linear derivations on $C^*$-algebras
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 1055-1065
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It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^nu)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ of real rank zero into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^n u)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all $u \in \lbrace v\in \mathcal A\mid v=v^*\hspace{5.0pt}\text{and}\hspace{5.0pt}v\hspace{5.0pt}\text{is} \text{invertible}\rbrace $, all $y\in \mathcal A$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^*$-algebras, and $\mathbb{C}$-linear $*$-derivations on unital $C^*$-algebras.
It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^nu)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ of real rank zero into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^n u)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all $u \in \lbrace v\in \mathcal A\mid v=v^*\hspace{5.0pt}\text{and}\hspace{5.0pt}v\hspace{5.0pt}\text{is} \text{invertible}\rbrace $, all $y\in \mathcal A$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^*$-algebras, and $\mathbb{C}$-linear $*$-derivations on unital $C^*$-algebras.
Classification : 39B52, 39B82, 46L05, 47B48
Keywords: $C^*$-algebra homomorphism; $C^*$-algebra; real rank zero; $\mathbb{C}$-linear $*$-derivation; stability 
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     title = {On homomorphisms between $C^*$-algebras and linear derivations on $C^*$-algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1055--1065},
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Park, Chun-Gil; Chu, Hahng-Yun; Park, Won-Gil; Wee, Hee-Jeong. On homomorphisms between $C^*$-algebras and linear derivations on $C^*$-algebras. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 1055-1065. http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a18/

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