The centre of the bidual of Fourier algebras (discrete groups)

TL;DR

This paper investigates the topological centre of the bidual of Fourier algebras for discrete groups, revealing differences based on group properties like amenability and non-abelian free groups.

Contribution

It characterizes the topological centre of the bidual of Fourier algebras for various classes of discrete groups, highlighting new distinctions based on group structure.

Findings

For amenable groups, the topological centre equals the Fourier algebra.

For groups containing a non-abelian free group, the topological centre is larger than the Fourier algebra.

Radial functions in the Fourier algebra are Arens regular.

Abstract

For a discrete group G with Fourier algebra A(G), we study the topological centre $Z_t$ of the bidual. If G is amenable, then $Z_t$ = A(G). But if G contains a non-abelian free group $F_r$, we show that $Z_t$ is strictly larger than A(G). Furthermore, it is shown that the subalgebra of radial functions in A(G) is Arens regular.