Geodesics of projections in von Neumann algebras

Abstract

Let A {\mathcal {A}} be a von Neumann algebra and P A {\mathcal {P}}_{\mathcal {A}} the manifold of projections in A {\mathcal {A}} . There is a natural linear connection in P A {\mathcal {P}}_{\mathcal {A}} , which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of C n \mathbb {C}^n . In this paper we show that two projections p , q p,q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A {\mathcal {A}} ), if and only if p ∧ q ⊥ ∼ p ⊥ ∧ q , \begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*} where ∼ \sim 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 p\wedge q^\perp = p^\perp \wedge q=0 . If A {\mathcal {A}} is a finite factor, any pair of projections in the same connected component of P A {\mathcal {P}}_{\mathcal {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 N ⊂ M {\mathcal {N}}\subset {\mathcal {M}} are II 1 _1 factors with finite index [ M : N ] = t − 1 [{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1} , then the geodesic distance d ( e N , e M ) d(e_{\mathcal {N}},e_{\mathcal {M}}) between the induced projections e N e_{\mathcal {N}} and e M e_{\mathcal {M}} is d ( e N , e M ) = arccos ⁡ ( t 1 / 2 ) d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2}) .