On orthomorphisms between von Neumann preduals and a problem of Araki
type: Item
instance of: scholarly article
title: On orthomorphisms between von Neumann preduals and a problem of Araki
zbMATH Open document ID: 0789.46049
author: J. D. Maitland Wright
author: Leslie J. Bunce
published in: Pacific Journal of Mathematics
publication date: 1994-06-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L70
zbMATH DE Number: 509104
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: orthogonal decomposition of Hermitian functionals
zbMATH Keywords: o.d. homomorphism
zbMATH Keywords: \(*\) Jordan homomorphism
zbMATH Keywords: preduals
MaRDI profile type: Publication
full work available at URL: pjm.1993.158.265
OpenAlex ID: W1965838692
DOI: 10.2140/PJM.1993.158.265
description: scientific article; zbMATH DE number 509104