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Weak Riemannian manifolds from finite index subfactors

Content Provider	Springer Nature Link
Author	Larotonda, Gabriel Andruchow, Esteban
Copyright Year	2008
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 $${\mathcal{O}({\sf p})=\{u\,{\sf p} u^{\ast} : u \in U_M\}}$$ of the Jones projection $${{\sf p}}$$ of the inclusion. We endow $${\mathcal{O}({\sf p})}$$ 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, $${\mathcal{O}({\sf p})}$$ is a weak Riemannian manifold. We show that $${\mathcal{O}({\sf p})}$$ 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 p 1 of $${\mathcal{O}({\sf p})}$$ , there is a ball $${\{q \in \mathcal{O}({\sf p}) : \|\|q - p_{1}\|\| < {r}\}}$$ (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 smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion $${\mathcal{O}({\sf p})\subset \mathcal{P}(M_1)}$$ , where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and $${\sf p}$$ .
Ending Page	232
Page Count	20
Starting Page	213
File Format	PDF
ISSN	0232704X
e-ISSN	15729060
Journal	Annals of Global Analysis and Geometry
Issue Number	3
Volume Number	34
Language	English
Publisher	Springer Netherlands
Publisher Date	2008-02-15
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Short geodesic Jones’ projection Totally geodesic submanifold Homogeneous manifolds Mathematical and Computational Physics Trace quadratic norm Finite index inclusion Riemannian, Finsler and other geometric structures Levi-Civita connection von Neumann II 1 subfactor Geometry General theory of von Neumann algebras Statistics for Business/Economics/Mathematical Finance/Insurance Geodesics Analysis Group Theory and Generalizations Homogeneous space Riemannian submanifold
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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