Mind the GAP: Amenability Constants and Arens Regularity of Fourier Algebras

Abstract

This thesis aims to investigate properties of algebras related to the Fourier algebra $A(G)$ and the Fourier-Stieltjes algebra $B(G)$, where $G$ is a locally compact group.

For a Banach algebra $\cA$ there are two natural multiplication operations on the double dual $\cA^{**}$ introduced by Arens in 1971, and if these operations agree then the algebra $\cA$ is said to be Arens regular. We study Arens regularity of the closures of $A(G)$ in the multiplier and completely bounded multiplier norms, denoted $A_M(G)$ and $A_{cb}(G)$ respectively. We prove that if a nonzero closed ideal in $A_M(G)$ or $A_{cb}(G)$ is Arens regular then $G$ must be a discrete group.

Amenable Banach algebras were first studied by B.E. Johnson in 1972. For an amenable Banach algebra $\cA$ we can consider its amenability constant $AM(\cA) \geq 1$. We are particularly interested in collections of amenable Banach algebras for which there exists a constant $\lambda > 1$ such that the values in the interval $(1,\lambda)$ cannot be attained as amenability constants. If $G$ is a compact group, then the central Fourier algebra is defined as $ZA(G) = ZL^1(G) \cap A(G)$ and endowed with the $A(G)$ norm. We study the amenability constant theory of $ZA(G)$ when $G$ is a finite group.