* - homomorphism between Banach *-algebra

Question

I want to prove a homomorphism of Banach * - algebra is a *- homomorphism if it maps self-adjoint element to self-adjoint element. I know every element is a unique sum of two self-adjiont elements. thanks for any hints.

Answer 1

' every element is a unique sum of two self-adjoint elements' is false. The correct statement is ' any element $x$ can be expressed uniquely as $x_1+ix_2$ where $x_1$ and $x_2$ are self-adjoint. Let $f$ be the given homomorphism. It is given that $(f(x))^{*} =f(x)$ whenever $x^{*}=x$. Write any element $x$ as $x_1+ix_2$ where $x_1$ and $x_2$ are self-adjoint. Then $(f(x))^{*} =(f(x_1+ix_2))^{*} =(u+iv)^{*}$ where $u=f(x_1)$ and $v=f(x_2)$. By hypothesis $u$ and $v$ are self-adjoint so $(f(x))^{*} =u^{*}-iv^{*}=u-iv=f(x_1)-if(x_2)=f(x_1-ix_2)=f(x^{*})$. Hence $f$ is a $*$ homomorphism.