Interplay between harmonic analysis and operator algebras

我的研究领域是谐波分析，数学的一个分支，涵盖或涉及广泛的学科，在纯数学和应用数学以及工程，如傅立叶和小波分析，信号处理，表示论和算子代数。我研究非阿贝尔群和某些典型的代数与他们有关。我的目标是在更深的层次上研究这些代数的结构和它们之间的相互作用，以及基础群的酉表示。 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。 与M。Wiersma，我们证明了长期存在的猜想，厄米特局部紧群是顺从的。许多有趣的方向，从这个证明，是值得追求的。例如，众所周知，厄米特的概念与群的增长密切相关，因此没有已知的厄米特离散群具有指数增长的例子。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的，甚至更好，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的增长性质联系起来。 在另一个方向上，我们还研究了奇异C *-代数，证明了具有快速衰减的Haagerup性质或Kuntz-Stein性质的非顺从群具有不可数的奇异C *-代数，大大推广了已有的结果.我的研究项目之一是适当地将这些结果扩展到更广泛的一类非顺从群，这样我就可以生成更多的奇异C *-代数。 我还研究了与傅立叶代数有关的各种性质。连通非交换群的Fourier代数不是弱顺从的，这一点已被证明了20多年。在一项联合工作中，我们证明了李群的这个猜想，我计划继续研究这个问题的其余情况。傅立叶代数的有限或无限相似度的存在，超出了迄今为止所知的，也是我的长期项目之一，我将进一步追求。 最后，在过去的十年中，我的研究计划的一部分集中在研究“局部行为”，即自反性和超自反性，上同调对象相关的这些代数和它们的关系与经典的算子理论。如果一个人看在研究算子代数的自反性和超自反性所使用的技术，一个人看到一些相似之处，并在同一时间，一些差异与工作的导子空间。我打算在这个方向上继续我的研究。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个理论传输到另一个理论。这可能有助于阐明这些领域中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.