Hilbert space effect algebras

Abstract

We introduce some natural numerical quantities which measure different kinds of correspondences between Hilbert space effect. We show that every bijective map of the space of all effects on the Hilbert space H which preserves any of these quantities is implemented by a unitary or antiunitary operator on H. In the las part we show that if an order preserving bijektive transformation of the Hilbert space effect algera also preserves the probability with respect to a fixed states, then it is an ortho-order automorphism. A similar result for the orthomodular lattice of all sharp effects (i.e..projektions) is also presented.