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Banach代数之间的相互作用，通常是代数的结构是内在相关的属性和动力学的基础组。所有我能看到自己积极参与的研究都建立在现有的研究成果之上。 大部分的洞察力将获得通过研究的结构傅立叶-Stieltjes，衡量代数，某些不动点子代数，并以类似的问题，围绕集团和傅立叶代数。 在许多阶段，有必要超越地方性的紧密团体。Fourier-Stieltjes（FS）代数我研究FS代数的新角度是我最近预印本[4]的Eberlein-de Leeuw-Glicksberg（EdLG）分解。这些分解产生于群如何作用于希尔伯特空间的动力学信息。他们在研究FS代数的算子顺从性方面取得了新的进展。此外，他们打开了大门，研究某些类的非局部紧群，这往往是自然产生的，即使在基础组是局部紧的。测度代数测度代数漂亮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，他们是对偶对象FS代数的方式推广庞特里亚金对偶。了解脊柱或EdLG分解的类似物将为更深入地了解其结构打开一扇大门。特别是，洞察力可以获得结构的幂等/投影和可逆的，也进入某些类别的非局部紧群。不动点子代数研究FS或测度代数的子代数的不动点的代数可以简化对代数的理解，但也显示了潜在的复杂性。它还邀请新的工具类到研究中，例如超群。关于顺从性的一些困难的开放性问题仍然存在，我希望直接攻击，或者通过寻找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通过一个C*-代数的任何适当有界的表示因子。我希望继续确定类的局部紧群的傅立叶代数享有这一财产。这些问题的类别为学生和博士后的培训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就帮助我们在量子信息理论等领域取得了领先地位，这在我的大学已经确立。

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.