Topological group actions and associated Banach algebras

我的研究集中在拓扑群和某些Banach代数类之间的相互作用，这些Banach代数通常是结构与基础群的性质和动力学有内在联系的代数。我认为自己积极参与的所有研究都建立在现有研究成果的基础上。许多洞察力将通过研究Fourier-Stieltjes的结构、度量代数、某些不动点子代数以及群和Fourier代数的相似性问题来获得。在许多阶段，有必要超越局部紧群。Fourier-Stieltjes(FS)代数我研究FS代数的新角度是我最近预印的Eberlein-de Leeuw-Glicksberg(EdLG)分解[4]。这些分解产生于关于群如何作用于希尔伯特空间的动力学信息。他们在考察FS代数的算子顺从性方面取得了新的进展。此外，它们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产生的，即使基础群是局部紧的。度量代数度量代数很好地编码了许多关于局部紧群的代数和拓扑信息，但它们的复杂性令人惊叹和敬畏。我的观点是，它们是FS代数的对偶对象，在某种程度上推广了Pontryagin对偶。了解脊椎，或EDLG分解的类似物，将打开一扇门，更深入地了解它们的结构。特别是，可以洞察幂等/投影和可逆的结构，也可以洞察某些类的非局部紧群。不动点子代数研究FS或测度子代数的不动点代数可以简化对该代数的理解，但也能揭示潜在的复杂性。它还邀请新的工具类别进入研究，例如超群。关于顺从性的一些棘手的公开问题仍然存在，我希望直接攻击这些问题，或者通过查看类似的问题来寻找线索。相似问题给定一个群(即它的L^1-代数)或一个傅立叶代数，我们希望了解在什么情况下任何适当有界的表示因子通过一个C*-代数。我希望继续确定傅立叶代数具有这一性质的局部紧群的类。这些问题课为培养学生和博士后提供了许多不同难度的切入点。加拿大在非对易调和分析方面保持着丰富的研究传统，它与我们擅长的算子代数等相关领域相联系。在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就帮助我们在量子信息论等领域取得了领先地位，这在我的大学已经很成熟了。

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.