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

TL;DR

This paper explores the geometric structure of the positive invertible elements in a finite von Neumann algebra, establishing properties like geodesic shortest paths, convexity, and a matrix-like factorization theorem.

Contribution

It introduces a Riemannian metric on the positive cone of a von Neumann algebra and proves geometric and algebraic properties analogous to matrix symmetric spaces.

Findings

Geodesics are shortest paths in the metric space.

The geodesic distance is convex.

A factorization theorem similar to Iwasawa decomposition is established.

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