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

Abstract

‎Let $A$ be a $C^{*}$ algebra‎, ‎$T‎: ‎Arightarrow A$ be a linear map which satisfies the functional equation $T(x)T(y)=T^{2}(xy),;;T(x^{*})=T(x)^{*} $‎. ‎We prove that under each of the following conditions‎, ‎$T$ must be the trivial map $T(x)=lambda x$ for some $lambda in mathbb{R}$: ‎‎ <br />‎i) $A$ is a simple $C^{*}$-algebra‎. <br />‎ii) $A$ 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. <br />‎iii) $A=B(H)$ where $H‎$‎ is a separable Hilbert space‎. <br />‎For a given field $F$‎, ‎we consider a similar functional equation {$ T(x)T(y) =T^{2}(xy), T(x^{tr})=T(x)^{tr}, $} 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 $nin mathbb{N}$ if and only if $F$ is a formally real field‎.