Positive linear maps on C⁎-algebras and rigid functions

Let a linear map Φ between two unital C⁎-algebras be positive and unital. Kadison showed that if f(t)=|t| and Φ(f(X))=f(Φ(X)) for all selfadjoint operators X, then Φ(X2)=Φ(X)2 for all selfadjoint operators X, that is, Φ is a C⁎-homomorphism. Choi proved this fact for an operator convex function f, and then conjectured that this fact would hold for a non-affine continuous function f. We shall prove a refinement of his conjecture. Petz has further proved that if f(Φ(A))=Φ(f(A)) for a non-affine operator convex function f and a fixed A, then Φ(A2)=Φ(A)2. Arveson called such a function f a rigid function. We shall directly show power functions tr are rigid functions on (0,∞) if r≠0,r≠1.