On orthomorphisms between von Neumann preduals and a problem of Araki
Let \(M\) and \(N\) be von Neumann algebras and let \(\phi: M_ *\to N_ *\) be a continuous linear map. If \(\phi\) is positive and preserves the orthogonal decomposition of Hermitian functionals then \(\phi\) is said to be an o.d. homomorphism. If \(\phi\) is a bijective o.d. homomorphism then it is said to be an o.d. isomorphism. The main results in the paper are Theorem. If \(\phi: M_ *\to N_ *\) is an o.d. homomorphism between the preduals of von Neumann algebras \(M\) and \(N\), then \(\phi^*\pi= z\cdot\text{id}_ M\), where \(\pi: M\to N\) is a \(*\) Jordan homomorphism and \(z\in Z(M)_ +\). Corollary. A continuous linear map \(\phi: M_ *\to N_ *\) between the preduals of von Neumann algebras \(M\) and \(N\) is an o.d. isomorphism if and only if \(\phi^*= z\pi\), where \(\pi: N\to M\) is a surjective \(*\) Jordan isomorphism and \(z\) is an invertible element in \(Z(M)_ +\). This answers, affirmatively, a problem of \textit{H. Araki} [Pac. J. Math. 137, No. 1, 1-13 (1989; Zbl 0696.46044)], where the above Corollary was proved in the case where one of \(M\) and \(N\) is Type I without Type I\(_ 2\) direct summands.
orthogonal decomposition of Hermitian functionals
scientific article; zbMATH DE number 509104