An extension of a characterization of the automorphisms of Hilbert space effect algebras

The aim of this paper is to show that if an order preserving bijective transformation of the Hilbert space effect algebra also preserves the probability with respect to a fixed pair of mixed states, then it is an ortho-order automorphism. A similar result for the orthomodular lattice of all sharp effects (i.e., projections) is also presented.