#### Vol. 12, No. 3, 2019

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

### Serina Camungol, Matthew Morison, Skylar Nicol and Ross Stokke

Vol. 12 (2019), No. 3, 491–502
##### 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.

##### Keywords
locally compact group, piecewise affine map, amenability, Fourier algebra
##### Mathematical Subject Classification 2010
Primary: 22D05, 43A22, 43A07, 43A30
Secondary: 20E99