@article {
author = {Taghavi, A.},
title = {On a functional equation for symmetric linear operators on $C^{*}$ algebras},
journal = {Bulletin of the Iranian Mathematical Society},
volume = {42},
number = {5},
pages = {1169-1177},
year = {2016},
publisher = {Iranian Mathematical Society (IMS)},
issn = {1017-060X},
eissn = {1735-8515},
doi = {},
abstract = {Let $A$ be a $C^{*}$ algebra, $T: A\rightarrow A$ 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 $n\in \mathbb{N}$ if and only if $F$ is a formally real field.},
keywords = {"Functional Equations","$C^{*}$ algebras"," Formally real fields"},
url = {http://bims.iranjournals.ir/article_872.html},
eprint = {http://bims.iranjournals.ir/article_872_fd7287eb7f1365d9156e9da3ccb25196.pdf}
}