Generalizing the Kantorovich Metric to Projection-Valued Measures

TL;DR

This paper extends the Kantorovich metric to projection-valued measures on a Hilbert space, establishing a complete metric space and applying it to fixed point results related to iterated function systems and Cuntz Algebras.

Contribution

It introduces a new generalized Kantorovich metric for projection-valued measures and explores its properties and applications, including a fixed point theorem.

Findings

The space of projection-valued measures is complete but not compact under the generalized metric.

The generalized metric has been previously defined in physics but is further developed here.

Application of the Contraction Mapping Theorem yields a fixed point related to iterated function systems.

Abstract

Given a compact metric space $X$, the collection of Borel probability measures on $X$ can be made into a compact metric space via the Kantorovich metric. We partially generalize this well known result to projection-valued measures. In particular, given a Hilbert space $\mathcal{H}$, we consider the collection of projection-valued measures from $X$ into the projections on $\mathcal{H}$. We show that this collection can be made into a complete and bounded metric space via a generalized Kantorovich metric. However, we add that this metric space is not compact, thereby identifying an important distinction from the classical setting. We have seen recently that this generalized metric has been previously defined by F. Werner in the setting of mathematical physics. To our knowledge, we develop new properties and applications of this metric. Indeed, we use the Contraction Mapping Theorem on this complete metric space of projection-valued measures to provide an alternative method for proving a fixed point result due to P. Jorgensen. This fixed point, which is a projection-valued measure, arises from an iterated function system on $X$, and is related to Cuntz Algebras.