Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X41Issue 7 (Special Issue)20151201Additive maps on C$^*$-algebras commuting with $|.|^k$ on normal elements8598724ENY.GuanDepartment of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.C.WangDepartment of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.J.HouDepartment of Mathematics, Taiyuan
University of Technology, Taiyuan 030024, P.R.
China.Journal Article20140908Let $\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.http://bims.iranjournals.ir/article_724_15cc6b48cc93f3f328eb35b3ec30359a.pdf