Weak Riemannian manifolds from finite index subfactors |
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Authors: | Esteban Andruchow Gabriel Larotonda |
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Institution: | (1) Instituto de Ciencias, Universidad Nacional de General Sarmiento, J. M. Gutierrez 1150, 1613 Los Polvorines, Argentina |
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Abstract: | Let N ⊂ M be a finite Jones’ index inclusion of II1 factors and denote by U
N
⊂ U
M
their unitary groups. In this article, we study the homogeneous space U
M
/U
N
, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit of the Jones projection of the inclusion. We endow 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, is a weak Riemannian manifold. We show that 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 p
1 of , there is a ball (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 smooth curve lying inside this ball. We also give an intrinsic
(algebraic) characterization of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and .
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Keywords: | Homogeneous space Short geodesic Levi-Civita connection Riemannian submanifold Totally geodesic submanifold Finite index inclusion von Neumann II 1 subfactor Jones’ projection Trace quadratic norm |
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