On a functional equation for symmetric linear operators on $C^{\ast }$ algebras

Abstract

Let $A$ be a $C^{\ast }$ algebra and $T:A\to A$ be a linear map which satisfies the functional equation $\{\begin{array}{l}T(x)T(y)=T^2(xy)\\ T(x^{\ast })=T(x{)}^{\ast }\end{array}$ We prove that under each of the following conditions, $T$ must be the trivial map $T(x)=\lambda x$ for some $\lambda \in \mathbb{R}:$\\ \begin{enumerate} \item $A$ is a simple $C^{\ast }$-algebra. \item $A$ is unital with trivial center and has a faithful trace such that each zero-trace element lies in the closure of the span of commutator elements. \item $A=B(H)$ where H is a separable Hilbert space. \end{enumerate} For a given field $F$, we consider a similar functional equation $$\{\begin{array}{l}T(x)T(y)=T^2(xy)\\ T(x^{tr})=T(x{)}^{tr}\end{array}$$ where $T$ is a linear map on $M_n(F)$ and "tr" is the transpose operator. We prove that this functional equation has trivial solution for all $n\in \mathbb{N}$ if and only if $F$ is a formally real field.

To appear in Bulletin of Iranian math Siciety

References in corpus (1)

• Zero product preservers of C*-algebras