Interplay between harmonic analysis and operator algebras

调和分析与算子代数之间的相互作用

• 批准号: RGPIN-2019-04420
• 负责人: Samei, Ebrahim
• 金额: $ 1.38万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=714467
• 关键词: Interplay; between; harmonic; analysis; operator

My research area is harmonic analysis, a branch of mathematics that covers or relates to a wide range of subjects in pure and applied mathematics as well as engineering such as Fourier and wavelet analysis, signal processing, representation theory and operator algebras. I study non-abelian groups and certain canonical algebras associated to them. My objective is to study at a deeper level the structure of these algebras and the interplay between them as well as the unitary representations of the underlying group. The connection between algebraic/analytic properties of group algebras or C*-algebras with the geometry of the underlying group is of great importance. Together with M. Wiersma, we proved the long-standing conjecture that Hermitian locally compact groups are amenable. Many interesting directions, arising from this proof, are worthy of pursuit. For example, it is well-known that the concept of Hermitian is closely related to the growth of the group so that there is no known example of a Hermitian discrete group with exponential growth. Hence a very interesting question is whether all discrete Hermitian groups are of subexponential growth, or even better, of polynomial growth. An affirmative answer connects a property related to harmonic analysis to the growth property of the group. On other direction, we also focused on exotic C*-algebras and showed that non-amenable groups with a "nice" Haagerup property possessing rapid decay or Kuntz-Stein property have uncountable many distinct exotic C*-algebras, greatly extending the known results so far. One of my research projects is to extend these results appropriately to a wider class of non-amenable groups so that I could generate many more exotic C*-algebras. I have also investigated various properties related to Fourier algebras. It has been conjectured for more than 20 years that the Fourier algebra of a connected nonabelian group fails to be weakly amenable. In a joint work, we proved this conjecture for Lie groups and I am planning to continue working on this question for the remaining cases. The existence of a finite or an infinite similarity degree for Fourier algebras, beyond what is known so far, is also one of my long-time projects that I will pursue further. Finally, for the last decade, a part of my research program has focused on studying “local behavior”, namely reflexivity and hyperreflexivity, of cohomological objects related to these algebras and their relation with the classical ones in operator theory. If one looks at the techniques used in studying reflexivity and hyperreflexivity of operator algebras, one sees some similarity to, and at the same time, some differences with the work on derivation spaces. I plan to pursue my research further in this direction. My ultimate goal is to make a bridge between these theories so that one could transfer information from one to another. This could help to shed light on some unsolved problem in these areas.

我的研究领域是调和分析，这是数学的一个分支，涵盖或涉及纯数学和应用数学以及工程领域的广泛学科，例如傅里叶和小波分析、信号处理、表示论和算子代数。我研究非阿贝尔群以及与它们相关的某些规范代数。我的目标是更深入地研究这些代数的结构、它们之间的相互作用以及基础群的统一表示。 群代数或 C* 代数的代数/解析性质与基础群的几何形状之间的联系非常重要。 我们与 M. Wiersma 一起证明了长期存在的猜想：厄米局部紧群是可行的。从这个证明中产生的许多有趣的方向都值得追求。例如，众所周知，厄米特群的概念与群的增长密切相关，因此没有已知的具有指数增长的厄米特离散群的例子。因此，一个非常有趣的问题是，是否所有离散厄米特群都是次指数增长，或者更好的是多项式增长。肯定的答案将与调和分析相关的属性与群的增长属性联系起来。 在其他方向上，我们还关注奇异的 C* 代数，并表明具有快速衰变的“良好”Haagerup 性质或 Kuntz-Stein 性质的不服从群具有不可数的许多独特的奇异 C* 代数，极大地扩展了迄今为止的已知结果。我的研究项目之一是将这些结果适当地扩展到更广泛的非服从群体，以便我可以生成更多奇异的 C* 代数。 我还研究了与傅立叶代数相关的各种性质。 20 多年来，人们一直猜测连通非阿贝尔群的傅立叶代数不具备弱服从性。在一项联合工作中，我们证明了李群的这个猜想，我计划继续针对其余案例研究这个问题。傅立叶代数存在有限或无限相似度，超出了迄今为止已知的范围，也是我将进一步追求的长期项目之一。 最后，在过去的十年中，我的研究计划的一部分集中于研究与这些代数相关的上同调对象的“局部行为”，即自反性和超自反性，以及它们与算子理论中经典代数的关系。如果我们看一下用于研究算子代数的自反性和超自反性的技术，我们会发现它们与推导空间的工作有一些相似之处，同时也有一些差异。我计划在这个方向上进一步进行研究。我的最终目标是在这些理论之间架起一座桥梁，以便人们可以将信息从一种理论转移到另一种理论。这可能有助于阐明这些领域中一些未解决的问题。