Homogeneous manifolds from noncommutative measure spaces

Abstract

Let M be a finite von Neumann algebra with a faithful normal trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U<SUB>M</SUB> of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ‖x‖<SUB>p</SUB> = τ (|x|<SUP>p</SUP>)<SUP>1/p</SUP>, p ≥ 1. The main results include the following. The unitary group carries on a rectifiable distance d<SUB>p</SUB> induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance over d<SUB>p</SUB> that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d<SUB>O, p</SUB>. We prove that the distances over d<SUB>p</SUB> and d<SUB>O, p</SUB> coincide. Based on this fact, we show that the metric space (O, d<SUB>p</SUB>) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of U<SUB>M</SUB> with the p-norm.