Positive linear maps on C⁎-algebras and rigid functions

Abstract 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 ( X 2 ) = 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 ( A 2 ) = Φ ( A ) 2 . Arveson called such a function f a rigid function . We shall directly show power functions t r are rigid functions on ( 0 , ∞ ) if r ≠ 0 , r ≠ 1 .