Nonpositive curvature in \(p\)-Schatten class
The cone \({\mathcal P}=M_n(\mathbb C)^+\) of positive invertible matrices, when given the metric induced by the trace of the algebra, and the action of the full linear group induced by the identification \(M_n(\mathbb C)^+\simeq GL(n,\mathbb C)/U(n,\mathbb C)\) (here \(U(n,\mathbb C)\) stands for the group of unitary matrices) is a Riemannian manifold of nonpositive curvature. This positive cone -- a homogeneous space of the full linear group -- is a universal model for (finite dimensional, symmetric) Riemannian manifolds of noncompact type [see \textit{P. B. Eberlein}, Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. Chicago, IL: The University of Chicago Press (1996; Zbl 0883.53003)]. Finsler norms can be introduced on \({\mathcal P}\), usually by ways of an unitarily invariant norm on the algebra \(M_n(\mathbb C)\). What is remarkable is that, if one replaces the Riemannian condition of nonpositive curvature (usually defined via the curvature tensor) in \({\mathcal P}\) by other equivalent notions such as inequalities for comparison triangles (à la Alexandrov or Busemann), or inequalities for the differential of the exponential map (see McAlpin's PhD thesis), then this cone turns out to be nonpositively curved for any unitarily invariant norm. Generalizing the classical setting, one can also consider positive cones of operator algebras, without dimension restrictions (\(C^*\)-algebras, von Neumann algebras). In this paper, the author studies the positive cone \(\Delta_p\) of the operator algebra that is obtained by adjoining the unit to the ideal of compact \(p\)-Schatten operators. The main results of the paper deal with the consequences of the \(p\)-uniform convexity [see \textit{K. Ball, E. A. Carlen} and \textit{E. H. Lieb}, Invent. Math. 115, No.~3, 463--482 (1994; Zbl 0803.47037)] of the tangent norms in this setting, which lead to strong convexity conditions on the geodesic distance in \(\Delta_p\), and in particular results on best approximation from convex submanifolds in \(\Delta_p\).
scientific article; zbMATH DE number 5566265
scientific article