On the Banach--Stone problem for \(L^p\)-spaces (Q2493627)

For a \(\sigma\)-finite measure space \((X,{\mathcal B},\mu)\), a well-known theorem of Lamperti describes surjective isometries \(T\) of \(L^p\) (\(p \neq 2\)) spaces in terms of regular set homomorphisms \(\Phi\) of the measure space. In particular, \(T(\chi_{B})= h \chi_{\Phi(B)}\) for \(B \in {\mathcal B}\). If the measure space also satisfies the Sikorski property, i.\,e., \(\Phi\) is induced by a measurable point-valued map \(\phi\), the authors show that \(T(f) = h (f \circ \phi)\). In other words, \(T\) is given by a composition operator, similar to the way the well-known Banach--Stone theorem describes isometries of \(C_0(X)\)-spaces.