Abstract

In this talk, after giving a brief background on the classical Beurling algebras, we introduce the class of Beurling-Fourier algebras on locally compact groups and show that they are non-commutative analogs of classical Beurling algebras. We obtain various results with regard to the operator amenability and Arens regularity of Beurling-Fourier algebras on compact groups and show that they behave very similarly to the classical Beurling algebras of discrete groups. We then apply our results to study explicitly the Beurling-Fourier algebras on $SU(2)$, the $2\times 2$ unitary group. We demonstrate that Beurling-Fourier algebras are closely connected to the amenability of the Fourier algebra of $SU(2)$. Another major consequence of our results is that our investigation allows us to construct families of unital infinite-dimensional closed subalgebras of the Fourier algebra of certain products of $SU(2)$ which are completely isomorphic to some operator algebras. If time permits, we will also investigate Beurling-Fourier algebras on $SU(n)$ and on Heisenberg groups.



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