Author/Authors :
Taghavi A. نويسنده Faculty of Mathematics and Computer Science, Damghan University, Damghan, Iran.
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 $lambda in mathbb{R}$: i) $A$ is a simple $C^{*}$-algebra. 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. iii) $A=B(H)$ where $H$ is a separable Hilbert space. 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.
