Weak Riemannian manifolds from finite index subfactors

Abstract

Let N ⊂ M be a finite Jones' index inclusion of II1 factors and denote by UN ⊂ UM their unitary groups. In this article, we study the homogeneous space UM/UN, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit O(p) = {u p u* : u ∈ UM} of the Jones projection p of the inclusion. We endow O(p) with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete); therefore, O(p) is a weak Riemannian manifold. We show that O(p) enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them are metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point p1 of O(p), there is a ball {q ∈ O(p) : ||q - p1|| < r} (of uniform radius r) of the usual norm of M, such that any point p2 in the ball is joined to p1 by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion O(p) ⊂ P(M1), where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and p.