On homomorphisms of an orthogonally decomposable Hilbert space (Q1086816)

Let M be a von Neumann algebra on a Hilbert space H with a cyclic separating vector. \(H^ J\), the real part of H, is then an ordered Hilbert space with respect to the natural cone \(H^+\). On this space, we investigate the properties of the non-commutative counterpart of the lattice homomorphism, which is called an o.d. homomorphism and is defined as a continuous linear operator \(\phi\) on H such that \(\phi (H^+)\subset H^+\) and \((\phi (\xi),\phi (\eta))=0\) whenever \(\xi \in H^+\), \(\eta \in H^+\), and \((\xi,\eta)=0\). Firstly a characterization of such operators is given. Secondly, we discuss the duals of o.d. homomorphisms and when they are normal. Thirdly, we consider semigroups of o.d. homomorphisms.