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Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra
Author(s) -
Andruchow E.,
Larotonda G.
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706022848
Subject(s) - mathematics , geodesic , invertible matrix , von neumann algebra , pure mathematics , trace (psycholinguistics) , metric (unit) , projection (relational algebra) , algebra over a field , combinatorics , mathematical analysis , von neumann architecture , linguistics , philosophy , operations management , algorithm , economics
In this paper we study the metric geometry of the space Σ of positive invertible elements of a von Neumann algebra 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 × 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 a suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained via a nonlinear orthogonal projection Π M : Σ → M , a map which turns out to be contractive for the geodesic distance.