  In this paper we study the metric geometry of the space $\Sigma$ of positive
invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite,
normal and faithful tracial state $\tau$. The trace induces an incomplete
Riemannian metric $<x,y>_a=\tau (ya^{-1}xa^{-1})$, and though the techniques
involved are quite different, the situation here resembles in many relevant
aspects that of the $n\times n$ matrices when they are regarded as a symmetric
space. For instance we prove that geodesics are the shortest paths for the
metric induced, and that the geodesic distance is a convex function; we give an
intrinsic (algebraic) characterization of the geodesically convex submanifolds
$M$ of $\Sigma$, and under suitable hypothesis we prove a factorization theorem
for elements in the algebra that resembles the Iwasawa decomposition for
matrices. This factorization is obtained \textit{via} a nonlinear orthogonal
projection $\Pi_M:\Sigma\to M$, a map which turns out to be contractive for the
geodesic distance.
