Characterizing sequential isomorphisms on Hilbert-space effect algebras

Abstract

Let ⋆ be any sequential product on the Hilbert-space effect algebra with , and be a bijective map. We show that if Φ satisfies Φ(A⋆B) = Φ(A)⋆Φ(B) for , then there is either a unitary or an anti-unitary operator U such that Φ(A) = UAU† for every . Let be a Borel function satisfying g(0) = 0, g(1) = 1 and let us define a binary operation ⋄g on by A⋄gB = A1/2 g(A)Bg(A)†A1/2, where T† denotes the conjugate of the operator T. We also show that a bijective map satisfies Φ(A⋄gB) = Φ(A)⋄gΦ(B) for if and only if there is either a unitary or an anti-unitary operator U such that Φ(A) = UAU† for every .