This is an announcement for the paper "On the geometry of von Neumann algebra preduals" by Miguel Martin and Yoshimichi Ueda.

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. 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.

Archive classification: math.OA

Remarks: 9 pages

Submitted from: ueda@math.kyushu-u.ac.jp

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1209.3391

or

http://arXiv.org/abs/1209.3391