Metric geometry of partial isometries in a finite von Neumann algebra

Abstract

We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that (Ip,dg) is a complete metric space, where dg is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).