Grant: Discovery Grants Program - Individual

My research program introduces a new mathematical structure, called Arveson system, and commences an exhaustive investigation of its properties in relation to the study of product systems and E0-semigroups. All these objects occur naturally in the theory of noncommutative dynamics, where they dictate the irreversible time-evolutions of quantum systems, and establish correlations between the anisotropy of physical time and the dynamical behavior of certain operator algebras.An Arveson system is a collection of involutive algebras A(r,s), indexed over all time intervals (r,s) with 0<r<s, which has the distinctive property that each algebra A(r,t) splits associatively into the tensor product of the algebras A(r,s) and A(s,t) in a certain way, for every 0<r<s<t. My research program is largely devoted to the classification and description of Arveson C*-systems and W*-systems. To investigate the main properties of these structures, I have developed new mathematical procedures that carry a distinctive quantum probabilistic character. Various concepts and techniques from the theory of stochastic processes, such as Levy processes, Levy-Ito decomposition, Brownian motions, stochastic integration and noisy systems, will be used, analyzed and studied in the framework of the theory of noncommutative dynamics, with the purpose of classifying and describing the structure of Arveson systems. These mathematical procedures have a great applicability potential, which will be exploited in order to solve some long-standing problems in the theory of noncommutative dynamics, such as classification of spectral C*-algebras associated with product systems of type III, or the problem of existence of E0-semigroups acting on any von Neumann algebra.The results of my research program aim to bring a new approach and a solid contribution to the study of noncommutative dynamics, which will surely strengthen Canada's international reputation in this active area of research.