Nonpositively curved metric in the positive cone of a finite von Neumann algebra

Abstract

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 $<x,y{>}_a=\tau (y{a}^{-1}x{a}^{-1})$, 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.

16 pages