Structure properties of non-selfadjoint operator algebras

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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.