Ph.D Thesis

Ph.D StudentShalit Orr
SubjectProduct Systems, Subproduct Systems and Dilation Theory of
Completely Positive Semigroups
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Baruch Solel


At the heart of this thesis lies the dilation theory of semigroups of completely positive maps acting on von Neumann algebras (henceforth CP-semigroups). Using a variety of functional analytic and operator algebraic tools, some of which are new, the existence - or nonexistence - of a *-endomorphic dilation to a CP-semigroup is established. When a *-endomorphic dilation exists this means that the CP-semigroup can be thought of as "part of" a semigroup of *-endomorphisms. The upshot is that a semigroup of *-endomorphsims is a special case of a CP-semigroup, and therefore is, at least in principle, better understood. The main results in the thesis regarding dilation theory of CP-semigroups are the following:

1. A dilation exists for two parameter semigroups satisfying the condition of "strong commutativity"

2.  Examples are provided showing that a dilation does not necessarily exist for three commuting CP maps, even when the maps are taken to be of arbitrarily small norm.

3. It is shown that the existence of a dilation is, in some cases, equivalent to a seemingly different problem, involving "subproduct systems".

The main strategy presented in this thesis is to approach the problem of dilating CP-semigroups through "subproduct systems" - an object that is formalized in this thesis. Building on previous works of Arveson, Muhly and Solel, to every CP-semigroup one can associate a subproduct system and a representation of that subproduct system. This reduces the problem of dilating a CP-semigroup to the problem of dilating a representation of a subproduct system. The thesis contains several new results regarding the latter problem as well.

After using subproduct systems to study dilation theory, the thesis focuses on subproduct systems in their own right. It is demonstrated that subproduct systems relate to, and are useful in the study of, operator algebras, noncommutative algebraic geometry, and multivariate operator theory.