On the preservation of properties by piecewise affine maps of locally compact groups

Abstract

As shown by Cohen (1960) and Ilie and Spronk (2005), for locally compact groups $G$ and $H$, there is a one-to-one correspondence between the completely bounded homomorphisms of their respective Fourier and Fourier–Stieltjes algebras $\phi :A\left(G\right)\to B\left(H\right)$ and piecewise affine continuous maps $\alpha :Y\subseteq H\to G$. Using elementary arguments, we show that several (locally compact) group-theoretic properties, including amenability, are preserved by certain continuous piecewise affine maps. We discuss these results in relation to Fourier algebra homomorphisms.