von Neumann algebra preduals satisfy the linear biholomorphic property

TL;DR

This paper proves that the preduals of infinite dimensional von Neumann algebras have a specific geometric property called the linear biholomorphic property, resolving a previously open problem.

Contribution

It establishes that the predual of any infinite dimensional von Neumann algebra has a zero symmetric part, solving a problem posed by Neal and Russo.

Findings

Preduals of infinite dimensional von Neumann algebras satisfy the linear biholomorphic property.

Symmetric part of the predual of JBW*-triples of rank >1 reduces to zero.

Addresses an open problem in the geometric theory of operator algebras.

Abstract

We prove that for every JBW$^*$-triple $E$ of rank $>1$, the symmetric part of its predual reduces to zero. Consequently, the predual of every infinite dimensional von Neumann algebra $A$ satisfies the linear biholomorphic property, that is, the symmetric part of $A_*$ is zero. This solves a problem posed by M. Neal and B. Russo in [Mathematica Scandinavica, to appear]