  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.
