Maps preserving strong products

Abstract

Let $\mathcal{𝒜}$ be an arbitrary $C^{\ast }$-algebra which contains a nontrivial projection $P_1$ and let $\phi :\mathcal{𝒜}\to \mathcal{𝒜}$ be a surjective map which satisfies

$$\phi \left(A_1\right){\bullet }_{\eta }\phi \left(A_2\right){\bullet }_{\eta }\cdots {\bullet }_{\eta }\phi \left(A_{n-1}\right){\bullet }_{\eta }\phi \left(P\right)=A_1{\bullet }_{\eta }{A}_2{\bullet }_{\eta }\cdots {\bullet }_{\eta }{A}_{n-1}{\bullet }_{\eta }P$$

for every $A_1,A_2,\dots ,A_{n-1}\in \mathcal{𝒜}$ $\left(n\ge 3\right)$, $P\in \left\{P_1,I-P_1\right\}$, and some $\eta \in \mathbf{C},$ such that $\left|\eta \right|=1$, $\eta \ne \pm 1$, and $A_1{\bullet }_{\eta }{A}_2{\bullet }_{\eta }\cdots {\bullet }_{\eta }{A}_{n-1}{\bullet }_{\eta }P$ is the Jordan multiple $\eta$-$\ast$-product where $A_1{\bullet }_{\eta }{A}_2=A_1{A}_2+\eta A_2{A}_1^{\ast }.$ We determine the concrete form of map $\phi$ on any arbitrary $C^{\ast }$-algebra. Also, if $n\ge 3$, $\eta =-1$ and $\mathcal{𝒜}$ is an arbitrary $\ast$-algebra (with unit $I$), over the complex field $\mathbf{C}$ which contains a nontrivial projection $P_1$, then $\phi$ is of the form $\phi \left(T\right)=T\phi \left(I\right)$ for all $T\in \mathcal{𝒜}$.