Exponential arcs in the manifold of vector states on a sigma-finite von Neumann algebra

TL;DR

This paper develops the concept of exponential arcs connecting vector states in sigma-finite von Neumann algebras, utilizing Tomita-Takesaki theory and non-commutative Radon-Nikodym theorems to analyze their properties.

Contribution

It introduces the notion of exponential arcs in the context of von Neumann algebras and demonstrates their differentiability and structural properties using advanced operator algebra techniques.

Findings

Exponential arcs are differentiable in the von Neumann algebra setting.

Parts of an exponential arc are themselves exponential arcs.

The approach applies to probability and quantum probability contexts.

Abstract

This paper introduces the notion of exponential arcs in Hilbert space and of exponential arcs connecting vector states on a sigma-finite von Neumann algebra in its standard representation. Results from Tomita-Takesaki theory form an essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem that involves positive operators affiliated with the commutant algebra. It is shown that exponential arcs are differentiable and that parts of an exponential arc are again exponential arcs. Special cases of probability theory and of quantum probability are used to illustrate the approach.