Structure properties of non-selfadjoint operator algebras

点击翻译按钮获取中文摘要

My mathematical research is in operator algebras. Operators and operator algebras are everywhere in the natural sciences since they are a combination of linear algebra and analysis. In particular, operator algebras and functional analysis form the backbone of modern quantum theory. In this way, my research is very relevant to a wider community than pure mathematics.***My proposed research will be made up of two main projects, crossed products of non-selfadjoint operator algebras and residually finite-dimensional operator algebras. Both of these projects into the structure of operator algebras are continuations of successful research I have conducted. ***First, I propose to continue my study of crossed products of non-selfadjoint operator algebras in collaboration with Elias Katsoulis of East Carolina University. A crossed product encodes the behaviour of a dynamical system. Elias and I developed this theory for non-selfadjoint operator algebras through three publications. This area has garnered some serious interest from the operator algebras community and it has, I believe, a rich theory still to be discovered.***A major open problem in crossed products is the Hao-Ng isomorphism problem for C*-correspondences. To every C*-correspondence one associates a canonical C*-algebra called the Cuntz-Pimsner algebra. Given a C*-correspondence and a certain action of a group the Hao-Ng isomorphism problem asks whether the crossed product of the Cuntz-Pimsner algebra of the C*-correspondence is isomorphic to the Cuntz-Pimsner algebra of the crossed product C*-correspondence. ***In the next five years I propose to resolve the Hao-Ng isomorphism problem. To achieve such a significant result will require the study of hyperrigidity and a constructive approach to dilation theory. ***The second major thrust of research that I am proposing is that of the study of residually finite-dimensional (RFD) operator algebras with Raphaël Clouâtre at the University of Manitoba. We have finished an initial study of these algebras which has opened up many further questions.***An operator algebra is called RFD if there is a family of finite-dimensional representations that recover the algebra. This property is a thriving research area in C*-algebras but has not been considered very much in the non-selfadjoint context. The study of representations is of significant importance to all areas of mathematics and finite-dimensional representations are particularly applicable to natural sciences and engineering. Basic questions remain open such as whether commutative operator algebras are RFD. I aim to dive deeply into such questions.*****