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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677016
• 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. ********

我的研究领域是谐波分析，这是数学的一个分支，涵盖或涉及纯数学和应用数学中的各种受试者以及傅立叶和小波分析，信号处理，表示理论和操作员代数等工程。我研究非亚洲群体以及与他们相关的某些规范代数。我的目标是在更深层次的层面上研究这些代数的结构以及它们之间的相互作用以及基础群体的统一表示。 *****与基础组的几何形状的组代数或C*代数的代数/分析性能之间的连接非常重要。我们与M. Wiersma一起证明了一个长期以来的概念，即Hermitian本地紧凑的群体是可正常的。由于这一证明而引起的许多有趣的方向值得追求。例如，众所周知，Hermitian的概念密切相关。对于该组的增长，因此没有已知的遗传学离散群体具有指数增长的例子。因此，一个非常有趣的问题是，所有离散的Hermitian群体是否具有次指数的增长，甚至更好地具有多项式增长。肯定的答案将与谐波分析有关的属性与小组的增长财产联系起来。 *****在其他方向上，我们还专注于异国情调的C* - 代数，并表明具有具有快速衰变或Kuntz-Stein属性的“不错的” haagerup特性的不符合的群体具有许多独特的外来的C* - 代数，迄今为止已知的结果很大。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不符合的群体，以便我可以产生更多的异国情调的C*-Elgebras。 *****我还研究了与傅立叶代数相关的各种属性。已经猜想了20多年，即连接的非亚伯群体的傅立叶代数无法稍微熟悉。在共同的工作中，我们为谎言团体提供了这个猜想，我计划继续在其余案件中处理这个问题。除迄今为止所知的傅立叶代数的有限或无限相似度的存在也是我将进一步追求的长期项目之一。 *****最终，在过去的十年中，我的研究计划的一部分集中在研究与这些代数有关的共同体对象及其与经营者理论中的经典相关的共同体对象的“本地行为”，即反思性和过度重新屈服。如果人们查看用于研究操作员代数的反射性和超反射性的技术，则可以看到与派生空间的工作有些差异，并且同时也有一些相似之处。我计划进一步朝这个方向进行研究。我的最终目标是在这些理论之间建立桥梁，以便可以将信息从一个理论转移到另一个理论。这可能有助于阐明这些领域的一些未解决的问题。 *****