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.

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