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We introduce the class of Beurling-Fourier algebras on locally compact groupsand show that they are non-commutative analogs of classical Beurling algebras.We obtain various results with regard to the operator amenability, operatorweak amenability and Arens regularity of Beurling-Fourier algebras on compactgroups and show that they behave very similarly to the classical Beurlingalgebras of discrete groups. We then apply our results to study explicitly theBeurling-Fourier algebras on SU(2), the 2 $\times$ 2 unitary group. Wedemonstrate that how Beurling-Fourier algebras are closely connected to theamenability of the Fourier algebra of SU(2). Another major consequence of ourresults is that our investigation allows us to construct families of unitalinfinite-dimensional closed Arens regular subalgebras of the Fourier algebra ofcertain products of SU(2).

Beurling-Fourier algebras, operator amenability and Arens regularity