Structure properties of non-selfadjoint operator algebras

我的数学研究方向是算子代数。算子和算子代数在自然科学中无处不在，因为它们是线性代数和分析的结合。特别是，算子代数和泛函分析形成了现代量子理论的支柱。这样，我的研究与比纯数学更广泛的社区非常相关。我的研究主要包括两个方面，非自伴算子代数的交叉积和剩余有限维算子代数。这两个项目的结构算子代数是延续的成功研究我已经进行。* 首先，我打算与东卡罗莱纳大学的Elias Katsoulis合作，继续研究非自伴算子代数的交叉积。交叉积编码了动力系统的行为。埃利亚斯和我通过三个出版物发展了非自伴算子代数的理论。这一领域已经引起了算子代数界的极大兴趣，我相信，它还有丰富的理论有待发现。交叉积中的一个主要的公开问题是C*-对应的Hao-Ng同构问题。对于每一个C*-对应，我们都将一个标准C*-代数称为Cuntz-Pimsner代数。给定一个C*-对应和一个群的某个作用，Hao-Ng同构问题是问C*-对应的Cuntz-Pimsner代数的交叉积是否同构于交叉积C*-对应的Cuntz-Pimsner代数。* 在接下来的五年里，我建议解决Hao-Ng同构问题。要达到这样一个重要的结果，将需要研究超刚性和膨胀理论的建设性方法。* 我提出的第二个主要研究方向是与马尼托巴大学的拉斐尔·克劳特雷一起研究剩余有限维（RFD）算子代数。我们已经完成了对这些代数的初步研究，这就提出了许多进一步的问题。一个算子代数被称为RFD，如果有一个有限维表示的家庭，恢复代数。这个性质是C*-代数中一个蓬勃发展的研究领域，但在非自伴背景下并没有得到太多的考虑。表示的研究是非常重要的所有领域的数学和有限维表示是特别适用于自然科学和工程。基本问题仍然悬而未决，如交换算子代数是否RFD。我的目标是深入研究这些问题。*

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.*****