Isometries on spaces of absolutely continuous functions in a noncompact framework (Q2306222)

Let \(X,Y\) be sets of real numbers with at least two points. This paper deals with a description of surjective isometries \(T\) of spaces of bounded, absolutely continuous functions on \(X,Y\), equipped with the norm \(\max\{\|f\|_{\infty}, bv(f)\}\), where \(bv(f)\) is the total variation of the function~\(f\). These results extend some earlier work of \textit{M. Hosseini} [J. Math. Anal. Appl. 463, No. 1, 386--397 (2018; Zbl 1439.46020)]. The proof proceeds along the expected way of showing that \(T(1)\) is a unimodular constant, \(\|T(f)\|_{\infty}=\|f\|_{\infty}\). Let \(X^\wedge\) denote the closure of \(X\) in its Stone-Čech compactification, then functions \(f\) admit unique extension to \(X^\wedge\). The description of \(T\) is in terms of monotonic homeomorphisms of \(Y^\wedge\), \(X^\wedge\) and multiplication by \(T(1)\).