Nonlinear centralizers with values in \(L_0\) (Q393205)

A function space is defined here as a linear subspace of \(L_0\), equipped with an \(F\)-norm which makes the embedding continuous. Such spaces carry a natural structure as modules over \(L_\infty\) (think of pointwise multiplication), and are considered in this category in all the following statements. Centralizers between function spaces form a class of quasilinear maps which arise in various situations; they include all maps which are continuous at the origin. In the case of maps into \(L_0\), it is proved that there are no others. One application is that, if \(L_\infty\) is dense in a function space \(X\) (in particular if \(X=L_0\)), then every twisted sum of \(L_0\) and \(X\) is trivial. Another is the existence of an Orlicz space \(L_N\) for which there is a non-trivial twisted sum of \(L_N\) and \(L_N^*\), even though the only homomorphism from \(L_N^*\) to \(L_N\) is \(0\).