Weak Riemannian manifolds from finite index subfactors
| Autor: | Esteban Andruchow, Gabriel Larotonda |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2008 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics 58B20 (Primary) 46L10 53C30 53C22 (Secondary) Geodesic Matemáticas RIEMANNIAN SUBMANIFOLD TOTALLY GEODESIC SUBMANIFOLD Levi-Civita connection Matemática Pura purl.org/becyt/ford/1 [https] symbols.namesake FOS: Mathematics Tangent space JONES' PROJECTION Ball (mathematics) LEVI-CIVITA CONNECTION Operator Algebras (math.OA) Riemannian submanifold Mathematics purl.org/becyt/ford/1.1 [https] Mathematics - Operator Algebras Riemannian manifold Submanifold VON NEUMANN II1 SUBFACTOR Differential Geometry (math.DG) Differential geometry FINITE INDEX INCLUSION TRACE QUADRATIC NORM SHORT GEODESIC symbols HOMOGENEOUS SPACE Geometry and Topology Mathematics::Differential Geometry Analysis CIENCIAS NATURALES Y EXACTAS |
| Zdroj: | CONICET Digital (CONICET) Consejo Nacional de Investigaciones Científicas y Técnicas instacron:CONICET |
| DOI: | 10.1007/s10455-008-9104-1 |
| Popis: | 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 a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $$ {\cal O}(p) =\{u p u^*: u\in U_M\} $$ of the Jones projection $p$ of the inclusion. We endow ${\cal 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 ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ 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 ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\ Comment: 19 pages |
| Databáze: | OpenAIRE |
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