Interplay between harmonic analysis and operator algebras

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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. ********