Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra

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 τ. The trace induces an incomplete Riemannian metric 〈 x,y 〉 a = τ ( ya −1 xa −1 ), and, though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the n × 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 a 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.