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构成了现代量子理论的支柱。通过这种方式，我的研究与更广泛的社区相关，而不是纯粹的数学。我提出的研究将由两个主要项目组成，非自伴算子代数的交叉积和剩余有限维算子代数。这两个关于算子代数结构的项目都是我所进行的成功研究的继续。首先，我建议与东卡罗来纳大学的Elias Katsoulis合作，继续我对非自伴算子代数的交叉积的研究。交叉积对动力系统的行为进行编码。Elias和我通过三篇论文为非自伴算子代数发展了这一理论。这个领域已经引起了算子代数界的一些认真的兴趣，我相信，它还有一个丰富的理论有待发现。交叉积中的一个主要公开问题是C*-对应的Hao-Ng同构问题。每个C*-对应都有一个规范的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代数的交叉积是否与交叉积C*-对应的Cuntz-Pimsner代数同构。在接下来的五年里，我建议解决Hao-Ng同构问题。要获得如此显著的结果，将需要研究超刚性和对膨胀理论的建设性方法。我提议的第二个主要研究主题是与马尼托巴大学的拉斐尔·L·克卢阿特一起研究剩余有限维算子代数。我们已经完成了对这些代数的初步研究，这打开了许多进一步的问题。一个算子代数称为RFD，如果存在恢复该代数的一族有限维表示。这一性质在C*-代数中是一个蓬勃发展的研究领域，但在非自伴背景下还没有被很好地考虑。表示法的研究对数学的各个领域都具有重要意义，而有限维表示法尤其适用于自然科学和工程学。诸如交换算子代数是否为RFD等基本问题仍然悬而未决。我的目标是深入探讨这些问题。

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.