Metric geometry of partial isometries in a finite von Neumann algebra

Abstract

We study the geometry of the setof 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 (I p , d g ) is a complete metric space, where d g is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).

Homogeneous manifolds from noncommutative measure spaces

“…Then Theorem 5.2 applies to this situation, and the curves δ(t) = e tz v with minimal symbol z are short among sufficiently short curves γ ∈ O. This example was studied in [1].…”

“…First we show thatḋ p ≤ d O,p . By Corollary 4.7, for each ǫ > 0, there exists a curve Γ ⊆ U M satisfying Γ(0) = uw 0 , Γ(1)…”

The set of partial isometries as a quotient Finsler space

“…Starting with Halmos and Mc Laughlin [14], who characterized the connected components. Later on, other papers appeared studying geometric or topological aspects of the set of partial isometries, for instance: [16], [18], [19], [1], [2], [4], [8], [9].…”

The set of partial isometries as a quotient Finsler space

“…Starting with Halmos and Mc Laughlin [12], who characterized the connected components. Later on, other papers appeared studying geometric or topological aspects of the set of partial isometries, for instance: [14], [16], [17], [1], [2], [4].…”