Date of Award
Doctor of Philosophy (PhD)
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 (Indiana Univ. Math. J. 30(5):713-747, 1981). We partially generalize this well known result to projection-valued measures. In particular, given a Hilbert space H, we consider the collection of projection-valued measures from X into the projections on 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 in 2004 (J. Quantum Inf. Comput. 4(6):546-562, 2004). 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 (see Adv. Appl. Math. 34(3):561-590, 2005; Operator Theory, Operator Algebras, and Applications, pp. 13-26, Am. Math. Soc., Providence, 2006). This fixed point, which is a projection-valued measure, arises from an iterated function system on X, and is related to Cuntz algebras. We conclude this document with a discussion of unitary representations of the Baumslag- Solitar group which arise from the Cantor set. We identify a family of partial isometries which can be used to construct the unitary operators which realize the representation of the Baumslag-Solitar group. These partial isometries satisfy relations similar to the Cuntz algebra relations.
Davison, Trubee Hodgman, "Generalizing the Kantorovich Metric to Projection-Valued Measures: With an Application to Iterated Function Systems" (2015). Mathematics Graduate Theses & Dissertations. 37.