Geodesics of projections in von Neumann algebras

Abstract

Let A be a von Neumann algebra and PA the manifold of projections in A. There is a natural linear connection in PA, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of Cn. In this paper we show that two projections p, q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A), if and only if p ? q? ? p? ? q, where ? stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ? q? = p? ? q = 0. If A is a finite factor, any pair of projections in the same connected component of PA (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones’ index theory for subfactors. For instance, it is shown that if N ?M are II1 factors with finite index [M : N ] = t?1, then the geodesic distance d(eN , eM) between the induced projections eN and eM is d(eN , eM) = arccos(t1/2).