Metric geometry of partial isometries in a finite von Neumann algebra

ESTEBAN ANDRUCHOW

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Año: 2008 vol. 337 p. 1226 - 1237

0022-247X

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∞p = 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 τM 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 ττ 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).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).(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). © 2007 Elsevier Inc. All rights reserved.2007 Elsevier Inc. All rights reserved. Keywords: Partial isometries; Projections; Geodesics; Finite von Neumann algebrasPartial isometries; Projections; Geodesics; Finite von Neumann algebras