Nonpositively curved metric in the positive cone of a finite von Neumann algebra
| dc.creator | Andruchow, Esteban | |
| dc.creator | Larotonda, Gabriel | |
| dc.date | 2008-08-13 | |
| dc.date.accessioned | 2026-07-07T09:56:24Z | |
| dc.date.available | 2026-07-07T09:56:24Z | |
| dc.description | In this paper we study the metric geometry of the space $Σ$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $τ$. The trace induces an incomplete Riemannian metric $<x,y>_a=τ(ya^{-1}xa^{-1})$, and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $Σ$, and under suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained \textit{via} a nonlinear orthogonal projection $Π_M:Σ\to M$, a map which turns out to be contractive for the geodesic distance. | |
| dc.description | 16 pages | |
| dc.identifier | https://arxiv.org/abs/0808.1774 | |
| dc.identifier | http://arxiv.org/abs/0808.1774 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/166968 | |
| dc.subject | Differential Geometry | |
| dc.subject | Operator Algebras | |
| dc.subject | 53C22, 58B20 (Primary) 46L45 (Secondary) | |
| dc.title | Nonpositively curved metric in the positive cone of a finite von Neumann algebra | |
| dc.type | text |