As shown by Cohen (1960) and Ilie and Spronk (2005), for locally compact groups and , there is a one-to-one correspondence between the completely bounded homomorphisms of their respective Fourier and Fourier–Stieltjes algebras and piecewise affine continuous maps . 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.
"On the preservation of properties by piecewise affine maps of locally compact groups." Involve 12 (3) 491 - 502, 2019. https://doi.org/10.2140/involve.2019.12.491