Artículo
Geodesics of projections in von neumann algebras
Fecha de publicación:
07/2021
Editorial:
American Mathematical Society
Revista:
Proceedings of the American Mathematical Society
ISSN:
0002-9939
e-ISSN:
1088-6826
Idioma:
Inglés
Tipo de recurso:
Artículo publicado
Clasificación temática:
Resumen
Let A be a von Neumann algebra and PA the manifold of projections in A. There is a natural linear connection in PA, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of Cn. In this paper we show that two projections p, q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A), if and only if p ∧ q⊥ ∼ p⊥ ∧ q, where ∼ stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ∧ q⊥ = p⊥ ∧ q = 0. If A is a finite factor, any pair of projections in the same connected component of PA (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones’ index theory for subfactors. For instance, it is shown that if N ⊂M are II1 factors with finite index [M : N ] = t−1, then the geodesic distance d(eN , eM) between the induced projections eN and eM is d(eN , eM) = arccos(t1/2).
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Articulos(IAM)
Articulos de INST.ARG.DE MATEMATICAS "ALBERTO CALDERON"
Articulos de INST.ARG.DE MATEMATICAS "ALBERTO CALDERON"
Citación
Andruchow, Esteban; Geodesics of projections in von neumann algebras; American Mathematical Society; Proceedings of the American Mathematical Society; 149; 10; 7-2021; 4501-4513
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