Gabriel Larotonda

Finsler geometry and actions of the p-Schatten unitary groups

Let p be an even positive integer and U p (H) the Banach-Lie group of unitary operators u which verify that u — 1 belongs to the p-Schatten ideal B p (H). Let O be a smooth manifold on which U p (H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of U p (H). Our main result establishes that for any pair of given initial conditions x ∈ O and X ∈ (TO) x there exists a curve 6(t) = e t z · x in O, with z a skew-hermitian element in the p-Schatten class, such that δ(0) = x and δ(0) = X, which remains minimal as long as t∥z∥ p < π/4. Moreover, δ is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups U p (H) and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu*: u ∈ U p (H)} of a self-adjoint operator A ∈ B(H).

Manifolds of semi-negative curvature

Fil: Conde, Cristian Marcelo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra

In this paper we study the metric geometry of the spaceof positive invertible elements of a von Neumann algebra A with a finite, normal and faithful tracial state �. The trace induces an incomplete Riemannian metric a= �(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 �, and under suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factoriza- tion is obtained via a nonlinear orthogonal projectionM : � → M, a map which turns out to be contractive for the geodesic distance. 1

SMOOTH PATHS OF CONDITIONAL EXPECTATIONS

Let A be a von Neumann algebra with a finite trace

Weak Riemannian manifolds from finite index subfactors

\tau

The left invariant metric in the general linear group

, represented in

Hopf-Rinow Theorem in the Sato Grassmannian ∗

H=L^2(A,\tau)

Norm Inequalities in Operator Ideals

, and let

The rectifiable distance in the unitary Fredholm group

B_t\subset A

Inequalities related to Bourin and Heinz means with a complex parameter

be sub-algebras, for