Positive linear operators in C^*-algebras
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It is shown that every almost positive linear mapping h : A→ B of a Banach *-algebra A to a Banach *-algebra B is a positive linear operator when h(rx) = rh(x) (r > 1) holds for all x ∈ 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^nu^*y) = h(2^nu)^*h(y) holds for all unitaries u ∈ A, all y ∈ A, and all n = 0, 1, 2, . . ., by using the Hyers-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.
Abstract
It is shown that every almost positive linear mapping h : A→ B of a Banach *-algebra A to a Banach *-algebra B is a positive linear operator when h(rx) = rh(x) (r > 1) holds for all x ∈ 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^nu^*y) = h(2^nu)^*h(y) holds for all unitaries u ∈ A, all y ∈ A, and all n = 0, 1, 2, . . ., by using the Hyers-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.
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