Spaces of nonpositive curvature arising from a finite algebra

TL;DR

This paper introduces a new class of noncomplete metric spaces modeled on finite von Neumann algebras with nonpositive curvature, exploring their geometric properties using metric geometry and functional analysis techniques.

Contribution

It presents novel examples of noncomplete nonpositively curved spaces derived from finite von Neumann algebras, expanding the understanding of such geometric structures.

Findings

Spaces are noncomplete but exhibit nonpositive curvature

Geodesic structures are analyzed using functional analysis methods

New insights into the geometry of algebraically modeled metric spaces

Abstract

In this paper we introduce a family of examples that can be regarded as spaces of nonpositive curvature, but with the distinct quality that they are not complete as metric spaces. This amounts to the fact that they are modelled on a finite von Neumann algebra, and the metrics introduced arise from the trace of the algebra. In spite of the noncompleteness of these manifolds, their geometry can be studied from the view-point of metric geometry, and several techniques derived from the functional analysis are applied to gain insight on their geodesic structure.