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

Abstract

Let $A$ be a $C^{*}$ algebra and $T: A\rightarrow A$ be a linear map which satisfies the functional equation $\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{*})=T(x)^{*} \end{cases}$ 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 $C^{*}$-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 $$\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{tr})=T(x)^{tr} \end{cases}$$ 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.