Geodesics of projections in von Neumann algebras

2020/11/03 by Andruchow, Esteban · 1 citation
#46L10 #53C22 #58B20 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2011.02013

Abstract

Let \cal A be a von Neumann algebra and \cal P_{\cal A} the manifold of projections in \cal A. There is a natural linear connection in \cal P_{\cal A}, 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 \cal A), if and only if p\wedge q^⊥∼ p^⊥\wedge 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\wedge q^⊥= p^⊥\wedge q=0. If \cal A is a finite factor, any pair of projections in the same connected component of \cal P_{\cal A} (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 \cal N⊂\cal M are \bf II₁ factors with finite index [\cal M:\cal N]=t^-1, then the geodesic distance d(e_{\cal N},e_{\cal M}) between the induced projections e_{\cal N} and e_{\cal M} is d(e_{\cal N},e_{\cal M})=\arccos(t^1/2).

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