Beurling-Fourier algebras, operator amenability and Arens regularity

TL;DR

This paper introduces Beurling-Fourier algebras on locally compact groups, explores their properties like operator amenability and Arens regularity, and applies these findings to specific groups such as SU(2).

Contribution

It defines a new class of non-commutative Beurling-Fourier algebras and analyzes their operator amenability, weak amenability, and Arens regularity, extending classical results to the non-commutative setting.

Findings

Beurling-Fourier algebras are non-commutative analogs of classical Beurling algebras.

They exhibit similar operator amenability and Arens regularity properties as classical Beurling algebras.

Explicit constructions of Arens regular subalgebras of Fourier algebras on products of SU(2).

Abstract

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, operator weak 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 how 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 Arens regular subalgebras of the Fourier algebra of certain products of SU(2).