Characterizing sequential isomorphisms on Hilbert-space effect algebras
Jinchuan Hou, Kan He, Xiaofei Qi · Journal of Physics A Mathematical and Theoretical · 2010
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 ◊ g B = A 1/2 g ( A ) Bg ( A )† A 1/2 , where T † denotes the conjugate of the operator T . We also show that a bijective map satisfies Φ( A ◊ g B ) = Φ( A )◊ g Φ( B ) for if and only if there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every .