On a functional equation for symmetric linear operators on C ∗ algebras

Let A be a C∗ algebraý, ýTý:ýA→A be a linear map which satisfies the functional equation T(x T(y)=T2(xy),T(x∗)=T(x)∗ý. ýWe prove that under each of the following conditionsý, ýT must be the trivial map T(x)=λx for some λ∈R: ýý ýi) A is a simple C∗-algebraý. ýii) A is unital with trivial center and has a faithful trace such that eachý ýzero-trace element lies in the closure of the span of commutator elementsý. ýiii) A=B(H) where Hýý is a separable Hilbert spaceý. ýFor a given field Fý, ýwe consider a similar functional equation {T(x)T(y)=T2(xy),T(xtr)= (x)tr,} where T is a linear map on Mn(F) andý ý"tr"ý ýis the transpose operatorý. ýWe prove that this functional equation has trivial solution for all n∈N if and only if F is a formally real fieldý.