%0 Journal Article
%T Additive maps on C$^*$-algebras commuting with $|.|^k$ on normal elements
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Guan, Y.
%A Wang, C.
%A Hou, J.
%D 2015
%\ 12/01/2015
%V 41
%N Issue 7 (Special Issue)
%P 85-98
%! Additive maps on C$^*$-algebras commuting with $|.|^k$ on normal elements
%K C$^*$-algebras
%K additive maps
%K Jordan homomorphism
%K *-homomorphism
%R
%X Let $\mathcal {A} $ and $\mathcal {B} $ be C$^*$-algebras. Assume that $\mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $\Phi:\mathcal{A} \to\mathcal{B}$ is an additive map preserving $|\cdot|^k$ for all normal elements; that is, $\Phi(|A|^k)=|\Phi(A)|^k $ for all normal elements $A\in\mathcal A$, $\Phi(I)$ is a projection, and there exists a positive number $c$ such that $\Phi(iI)\Phi(iI)^{*}\leq
c\Phi(I)\Phi(I)^{*}$, then $\Phi$ is the sum of a linear Jordan *-homomorphism and a conjugate-linear Jordan *-homomorphism. If, moreover, the map $\Phi$ commutes with $|.|^k$ on $\mathcal{A}$, then $\Phi$ is the sum of a linear *-homomorphism and a conjugate-linear *-homomorphism. In the case when $k \not=1$, the assumption $\Phi(I)$ being a projection can be deleted.
%U http://bims.iranjournals.ir/article_724_15cc6b48cc93f3f328eb35b3ec30359a.pdf