Spectral order on unbounded operators and their symmetries

The spectral order on positive unbounded operators affiliated with von Neumann algebras have been considered. The spectral order has physical meaning of comparing distribution functions of quantum observables and organizes the structure of unbounded positive operators into a complete lattice. In the previous investigation, we clarified the structure of canonical preservers of the spectral order relation in the bounded case. In the present paper, we have discussed new results on preservers of the spectral order for unbounded positive operators affiliated with von Neumann algebras. We proved earlier that any spectral automorphisms (bijection preserving the order in both directions) of the set of all positive unbounded operators acting on a Hilbert space is a composition of function calculus with a natural extension of projection lattice automorphism. Our investigation starts with observation that this does not hold if the underlying von Neumann algebras have a non-trivial center. However, we have shown that for any von Neumann algebra the following holds. The spectral automorphism preserves positive multiples of projections if and only if it is a composition of the function calculus given by a strictly increasing bijection of the positive part of the real line and an extension of projection lattice automorphism.