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The Kantorovich metric provides a way of measuring the distance between two Borel probability measures on a metric space. This metric has a broad range of applications from bioinformatics to image processing, and is commonly linked to the optimal transport problem in computer science. Noteworthy to this paper will be the role of the Kantorovich metric in the study of iterated function systems, which are families of contractive mappings on a complete metric space. When the underlying metric space is compact, it is well known that the space of Borel probability measures on this metric space, equipped with the Kantorovich metric, constitutes a compact, and thus complete metric space. In previous work, we generalized the Kantorovich metric to operator-valued measures for a compact underlying metric space, and applied this generalized metric to the setting of iterated function systems. We note that the work of P. Jorgensen, K. Shuman, and K. Kornelson provided the framework for our application to this setting. The situation when the underlying metric space is complete, but not necessarily compact, has been studied by A. Kravchenko. In this paper, we extend the results of Kravchenko to the generalized Kantorovich metric on operator-valued measures., Comment: The current version has been updated with small changes to match the final publication, Acta Applicandae Mathematicae (2018)

Given an iterated function system (IFS) on a complete and separable metric space $Y$, there exists a unique compact subset $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap. In previous work, we developed an alternative approach to proving the existence of this projection-valued measure. The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in. We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory. We conclude with a discussion of Naimark's dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric., Comment: The current version has been updated with small changes to match the final publication, Acta Applicandae Mathematicae (2018)

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., Comment: The new version of this article has been updated to correct an error (proof of Theorem 3.1 in the previous version). We would like to acknowledge Krystal Taylor for identifying the error. The new version has also been shortened from the previous version (removed Section 4) to match the final publication