On the preservation of properties by piecewise affine maps of locally compact groups (Q1731041)

Let $G, H$ be locally compact groups. Mixed piecewise affine mappings between (subsets of) $H$ and $G$ have been mainly studied through their connection with homomorphisms from $A(G)$, the Fourier algebra on $G$, into $B(H)$, the Fourier-Stieltjes algebra on $H$ (such as in [\textit{P. J. Cohen}, Amer. J. Math. 82, 213--226 (1960; Zbl 0099.25601)] and in [\textit{M. Ilie} and \textit{N. Spronk}, J. Funct. Anal. 225:2, 480--499 (2005; Zbl 1077.43004)]). In this paper, the author initiates the studies of these mappings on their own as a weaker type of morphisms between locally compact groups. Let $Y$ be a subset of $H$ and let $\alpha$ be a mapping $Y\to G$. Then $\alpha$ is called a \textit{piecewise affine mapping} if $Y=\bigcup_{j=1}^n Y_j$ where $Y_j$ are disjoint and each $Y_j$ is the difference of an open coset of $H$ by a finite union of other open cosets of $H$, and each restriction $\alpha|Y_j$ extends to a translation of a continuous group homomorphism from an open subgroup of $H$ into $G$. If some of the group homomorphisms are allowed to be replaced by group anti-homomorphisms, then $\alpha$ is called a \textit{mixed piecewise affine mapping}. The main questions that the authors address in this paper is which properties $\mathcal{P}$ of locally compact groups are preserved after passing through a mixed piecewise affine mapping. In particular, given a mixed piecewise affine mapping $\alpha:Y\subseteq H\to G$: \begin{itemize} \item[(a)] If $H$ has $\mathcal{P}$ and $\alpha(H)$ is dense in $G$, does $G$ have $\mathcal{P}$? \item[(b)] If $Y=H$ and $\alpha$ is injective (or proper), if $G$ has $\mathcal{P}$ does $H$ have $\mathcal{P}$? \end{itemize} The authors prove some general results that imply that (a) holds for the properties ``amenability'' and ``compactness'' and (b) holds for the properties ``amenability'', ``compactness'', and ``being an [IN]-group''.