POSITIVE LINEAR OPERATORS IN C*-ALGEBRAS

Abstract

It is shown that every almost positive linear mapping h : A 13 of a Banach *-algebra A to a Banach *-algebra 13 is a positive linear Operator when h(rx) = rh(x) (r > 1) holds for all x E A, and that every almost linear mapping h : A -> B of a unital C*-algebra A to a unital C*-algebra B is a positive linear operator when h(2(n)u*y) = h(2(n)u)* h(y) holds for all unitaries a is an element of A, all y is an element of A, and all n = 0,1,2, ... , by using the Flyers-Ulam-Rassias stability of functional equations. Under a more weak condition than the condition as given above, we prove that every almost linear mapping h : A -> B of a unital C*-algebra A to a unital C*-algebra B is a positive linear operator. It is applied to investigate states, center states and center-valued traces.