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，我的研究与一个比纯数学更广泛的社区非常相关。***我提出的研究将由两个主要项目组成，非自伴随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延续。***首先，我提议与东卡罗莱纳大学的Elias Katsoulis合作，继续研究非自伴随算子代数的交叉积。交叉积对动力系统的行为进行编码。Elias和我通过三篇论文发展了这个非自伴随算子代数的理论。这个领域已经引起了算子代数界的一些严肃的兴趣，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开放问题是C*-对应的Hao-Ng同构问题。对于每一个C*对应，都有一个称为康茨-皮姆斯纳代数的标准C*代数。给定一个C*对应和一个群的某种作用，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。***我建议在未来五年内解决Hao-Ng同构问题。要取得如此重要的结果，需要对超刚性进行研究，并对膨胀理论采取建设性的方法。***我提出的第二个主要研究方向是曼尼托巴大学的Raphaël clou<e:2>对剩余有限维算子代数的研究。我们已经完成了对这些代数的初步研究，并由此提出了许多进一步的问题。***如果有一组有限维表示可以恢复该代数，则称为RFD。这一性质在C*-代数中是一个蓬勃发展的研究领域，但在非自伴的情况下却没有得到很好的考虑。表征的研究对数学的所有领域都具有重要意义，有限维表征尤其适用于自然科学和工程。基本的问题仍然没有解决，比如交换算子代数是否为RFD。我打算深入研究这些问题。*****

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.*****