On Multi-Parameter CCR Flows [HBNI Th252]

Abstract

An E 0 -semigroup is a semigroup of normal, unital §-endomorphisms of B(H ), where H is an infinite-dimensional separable Hilbert space. Though initially considered only in the one-parameter context, multiparameter E 0 -semigroups have recently been studied and provide for an interesting area of research. The goal of this thesis is to further explore multiparameter CCR flows, which are a class of E 0 -semigroups, and to produce prime multiparameter CCR flows with non-trivial index.

In the first part of this thesis, we provide the preliminaries required to understand the subject. We also define the index of a spatial E 0 -semigroup over a closed, convex cone in Rd , extending the definition of Arveson’s index from the one-parameter theory. We prove that the index of a CCR flow is equal to the index of the associated isometric representation. We also provide a necessary and sufficient condition for when such a CCR flow will be of type I.

In the second part of this thesis, we study E 0 -semigroups indexed by Lie semi- groups. We show that the CCR functor is injective when considered in the context of Lie semigroups, thereby extending results from the one-parameter theory and also the case where the semigroup considered is a closed, convex cone in Rd . Next, we analyse a class of CCR flows that was previously considered over closed, convex cones by Arjunan, Srinivasan and Sundar. We show that, in the non-commutative case, the results obtained in the commutative case hold but partially. We also construct the first examples of prime CCR flows over closed convex cones that have index one. In the last part of our thesis, we induce isometric representations from discrete semigroups to continuous semigroups and use this to construct uncountably many examples of prime multiparameter CCR flows with any given index.