On orthomorphisms between von Neumann preduals and a problem of Araki

Abstract

A problem of Araki concerning the characterization of orthogo- nality preserving positive maps between preduals of von Neumann algebras is solved in a general setting. Introduction. In an interesting recent article, Araki (1) initiated the study of orthogonal decomposition preserving positive linear maps (o.d. homomorphisms) between preduals of von Neumann algebras. (See below for definitions.) Let M and N be von Neumann algebras and let φ: M* -» N* be a linear mapping. When either M or N is of Type I, with no direct summand of Type I2, Araki proved that φ is a bijective o.d. homo- morphism if, and only if, φ* = zπ where z is a positive invertible element of the centre of M and π:N -> M is a Jordan isomorphism. Araki posed the problem of establishing an analogous characteriza- tion when M and N were of Type II or Type III. Araki used delicate Radon-Nikodym methods which seem very dif- ficult to generalize to algebras which are not of Type I. However, by adopting a different approach, we are able to show, for arbitrary von Neumann algebras M and N, that if φ: M* —> N* is an o.d. homo- morphism then φ*π = z id^ where z is a positive central element of M and π is a Jordan * homomorphism, and we obtain a character- ization in these terms. If φ is an o.d. isomorphism, we find that z is invertible and that π is a Jordan * isomorphism. This proves that Araki's characterization of o.d. isomorphisms is valid for arbitrary von Neumann algebras M and N. 1. Preliminaries . Two positive linear functional p, τ in the pred- ual M* of a W* -algebra M are said to be orthogonal, written p _L τ, if the corresponding support projections s(p), s(τ) are orthogonal ele- ments in the algebra M. Every hermitian functional p in Af* admits a unique orthogonal decomposition p = ρ+-p- , where />+, p- € A/+ and p+ ± p-. On the other hand every hermitian element x in M has a unique orthogonal decomposition x = x+ = X-, where x+, x_ > 0 and x+ x_ = 0.