Interplay between harmonic analysis and operator algebras

我的研究领域是谐波分析，这是数学的一个分支，涵盖或涉及到纯数学和应用数学以及工程中的广泛学科，如傅里叶和小波分析、信号处理、表示理论和算子代数。我研究非阿贝尔群和与之相关的正则代数。我的目标是在更深层次上研究这些代数的结构和它们之间的相互作用，以及底层群的酉表示。群代数或C*-代数的代数/解析性质与基础群的几何之间的联系是非常重要的。我们与M. Wiersma一起证明了厄米局部紧群是可服从的长期猜想。从这一证明中产生了许多有趣的方向，值得探索。例如，众所周知，厄米离散群的概念与群的增长密切相关，因此没有已知的厄米离散群具有指数增长的例子。因此一个很有趣的问题是是否所有离散厄米群都是次指数增长，或者更好的是多项式增长。一个肯定的答案将与调和分析相关的性质与群体的生长性质联系起来。在另一个方向上，我们也关注了奇异C*-代数，并证明了具有快速衰变的“好”Haagerup性质或kunz - stein性质的非可调群具有无数不同的奇异C*-代数，极大地扩展了迄今为止已知的结果。我的一个研究项目是将这些结果适当地扩展到更广泛的不可适应群，这样我就可以产生更多奇特的C*-代数。我还研究了与傅里叶代数相关的各种性质。20多年来，人们一直推测连通非贝尔群的傅里叶代数不具有弱可服从性。在一项联合工作中，我们证明了李群的这个猜想，我计划继续在剩余的情况下研究这个问题。傅里叶代数的有限或无限相似度的存在，超出了目前已知的范围，也是我的长期项目之一，我将继续研究。最后，在过去的十年里，我的研究计划的一部分集中在研究与这些代数相关的上同调对象的“局部行为”，即反身性和超反身性，以及它们与算子理论中经典代数的关系。如果我们观察研究算子代数的自反性和超自反性所使用的技术，我们会发现它们与导数空间的研究既有相似之处，也有不同之处。我打算在这个方向上进行进一步的研究。我的最终目标是在这些理论之间架起一座桥梁，这样人们就可以将信息从一个理论传递到另一个理论。这可能有助于阐明这些领域的一些尚未解决的问题。

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.