  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.

Operator Algebras