Weak Riemannian Manifolds from Finite Index Subfactors

ESTEBAN ANDRUCHOW; GABRIEL LAROTONDA

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY

Springer

Año: 2008 vol. 34 p. 213 - 232

0232-704X

Let N subset M be a finite  Jones´ index inclusion of II_1 factors, and denote by U_N subset U_M their unitary groups. In this paper we study the homogeneous space U_M/U_N, which is an (infinite dimensional) differentiable manifold, diffeomorphic to the orbit o(p)={upu*: u \in U_M} 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, 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 p_1 of o(p), there is a ball {q in o(p): || q - p_1 ||<r} (of uniform radius r) of the usual norm of M, such that any point p_2 in the ball is joined to p_1 by a unique geodesic, which is shorter than any other piecewise smoothcurve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion o(p) subset P(M_1), where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and p.