C*-Envelopes and Other Themes in Operator Algebras

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This research project seeks to resolve various problems in operator algebra theory, an area of mathematical analysis. The general study of operator algebras began in the 1930’s in an effort to provide a mathematical formulation of particle physics and quantum mechanics, notably to model algebras of physical observables. Since then, this vibrant area of mathematics has become an independent discipline bringing valuable insight to other areas of mathematics (knot theory, dynamical systems, group theory), physics (statistical mechanics) and engineering (signal processing). The award will contribute to US workforce development through training of students at East Carolina University. There are three major areas of study in this project: the C*-envelope of an operator algebra, the structure theory of crossed products and isomorphism invariants for operator algebras. The methods employed in the study of these areas are that of Functional Analysis with a strong emphasis on modern dilation theory and representation theory of associative algebras. A particular feature of the project is the use, for the first time, of non-selfadjoint techniques in an attempt to resolve issues in the selfadjoint theory, including the Hao-Ng isomorphism problem and the description of co-universal C*-algebras for covariant representations of product systems. Conversely, this project seeks to incorporate in the toolkit of non-selfadjoint operator algebraists techniques which are standard in the selfadjoint world. For instance, classifying Arveson’s semicrossed products up to stable isomorphism relates directly to the problem of classification of crossed products via a novel use of non-selfadjoint Takai duality. Such a synergy has not been explored before and seems to open exciting possibilities for both the selfadjoint and non-selfadjoint theory of operator algebras. Towards this end, the PI intends to continue organizing seminars and workshops aimed at further facilitating collaboration between experts in these two fields while at the same time promoting the exposure of students and early-career mathematicians to this promising area of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.