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 Σ of positive invertible elements of a von Neumann algebra. A with a finite, normal and faithful tracial state T. The trace induces an incomplete Riemannian metric _a=T(ya^{-1}xa^{-1}), and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the n x 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 Σ, 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 via a nonlinear orthogonal projection π_M: Σ → M, a map which turns out to be contractive for the geodesic distance.