Topological group actions and associated Banach algebras

我的研究主要集中在拓扑群和某些类别的巴拿赫代数之间的相互作用，这些代数的结构通常与底层群的性质和动力学具有内在联系。我所从事的所有研究都是建立在现有研究成果的基础上的。通过对Fourier- stieltjes结构、测度代数、某些不动点子代数以及围绕群和Fourier代数的相似性问题的研究，将获得许多见解。在许多阶段，有必要超越地方性的紧密团体。我研究FS代数的新角度是我最近的预印本[4]中的Eberlein-de Leeuw-Glicksberg （EdLG）分解。这些分解来自于群如何作用于希尔伯特空间的动力学信息。它们为检验FS代数的算子适性提供了新的途径。此外，它们还为研究某些非局部紧群打开了大门，这些非局部紧群通常是自然出现的，即使底层群是局部紧的。测度代数测度代数很好地编码了许多关于局部紧群的代数和拓扑信息，但其复杂性令人生畏。我的观点是，它们是FS代数的对偶对象，以一种推广庞特里亚金对偶的方式。了解脊椎或EdLG分解的类似物将为更深入地了解它们的结构打开一扇门。特别地，可以对幂等/投影和可逆的结构，以及某些非局部紧群的类，有深入的了解。不动点子代数研究FS或测度代数的子代数不动点的代数可以简化对代数的理解，但也显示出潜在的复杂性。它还引入了新的工具类别，例如超群。关于适应性的一些难以解决的问题仍然存在，我要么希望直接攻击它们，要么通过观察类似的问题来寻求提示。给定一个群（即它的L^1代数）或一个傅立叶代数，我们希望理解在什么情况下任何合适的有界表示通过C*-代数因子。我希望继续确定傅里叶代数具有这种性质的局部紧群的种类。这类问题为培养学生和博士后提供了许多不同难度的切入点。加拿大在非交换调和分析方面保持着丰富的研究传统，这与我们擅长的算子代数等相关领域有关。这些领域的培训培养了许多优秀的研究人员，并通过人才和研究成果帮助我们在量子信息理论等领域奠定了领先地位，这在我的大学已经建立起来。

My research is focused on the interplay between topological groups and certain classes of Banach algebras, typically algebras whose structures are intrinsically related to the properties and dynamics of the underlying group. All of the research in which I can see myself active builds upon existing research accomplishments.  Much of the insight will be gained through the study of the structure of Fourier-Stieltjes, measure algebras, certain fixed point subalgebras, and to similarity problems around group and Fourier algebras.  At many stages, there is necessity to move beyond locally compact groups. Fourier-Stieltjes (FS) algebras My new angle for studying FS algebras are the Eberlein-de Leeuw-Glicksberg (EdLG) decompositions of my recent preprint [4]. These decompositions arise from information of the dynamics of how groups act on Hilbert spaces. They have given a new inroad in examining operator amenability of FS algebras. Furthermore, they open the door to studying certain classes of non-locally compact groups, which often arise naturally, even when the underlying group is locally compact. Measure algebras Measure algebras beautifully encode much algebraic and topological information about locally compact groups, but are magnificent and daunting in their complexity. My view is that they are dual objects to FS algebras in a manner generalizing Pontryagin duality. Understanding the spine, or the analogue of EdLG decompositions will open a door to learn more deeply about their structures. In particular, insight may be gained into structures of idempotent/projections and of invertibles, and also into certain classes of non-locally compact groups. Fixed point subalgebras The study of the algebra of fixed points of a subalgebra of an FS or measure algebra can simplify the understanding of the algebra, but also show underlying complexities. It also invites new classes of tools into the study, e.g. hypergroups. Some difficult open problems about amenability remain which I either wish to directly attack, or seek hints by looking at analogous problems. Similarity problems Given a group (i.e. its L^1-algebra) or a Fourier algebra, we wish to understand under what circumstances any suitably bounded representation factors through a C*-algebra. I wish to continue determining classes of locally compact groups for which Fourier algebras enjoy this property. The classes of problems give many entry points, of varying difficulty, for training of students and post-docs.  Canada maintains a rich tradition of research in non-commutative harmonic analysis, which links to related fields such as operator algebras, in which we are strong.  Training in these fields has produced many good researchers, and has helped seed, through talent and research accomplishments, our leading position in such fields as quantum information theory, which is well-established at my university.