ON A FUNCTIONAL EQUATION FOR SYMMETRIC LINEAR OPERATORS ON C* ALGEBRAS

Let A be a C∗ algebra‎, ‎T‎:‎A→A be a linear map which satisfies the functional equation T(x)T(y)=T2(xy),T(x∗)=T(x)∗‎. ‎We prove that under each of the following conditions‎, ‎T must be the trivial map T(x)=λx for some λ∈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)=T2(xy),T(xtr)=T(x)tr,} where T is a linear map on Mn(F) and‎ ‎ tr ‎ ‎is the transpose operator‎. ‎We prove that this functional equation has trivial solution for all n∈N if and only if F is a formally real field‎.