On the geometry of von Neumann algebra preduals

TL;DR

This paper investigates the geometric structure of von Neumann algebra preduals, identifying conditions under which certain norm equations are solvable, and relates these properties to the Daugavet property in the context of operator algebras.

Contribution

It characterizes when the predual of a von Neumann algebra satisfies a specific norm equation, linking this to the algebra's type decomposition and the Daugavet property.

Findings

The norm equation holds iff the algebra lacks type I and type III₁ summands.

The property is characterized by centrally symmetric curves in the unit sphere.

For diffuse algebras, the equation is solvable in ultraproducts, linking to the Daugavet property.

Abstract

Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual. We study when for every $\varphi\in M_\star$ there exists $\psi\in M_\star$ solving the equation $\|\varphi \pm \psi\|=\|\varphi\|=\|\psi\|$. This is the case when $M$ does not contain type I nor type III$_1$ factors as direct summands and it is false at least for the unique hyperfinite type III$_1$ factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $M_\star$ of length 4. An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.