Metric geometry of partial isometries in a finite von Neumann algebra
We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does n...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2008 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/19448 |
| Acceso en línea: | http://hdl.handle.net/11336/19448 |
| Access Level: | acceso abierto |
| Palabra clave: | Partial Isometry Finite Algebra Homogeneous Spaces https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that (Ip,dg) is a complete metric space, where dg is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves). |
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