Nonlinear centralizers with values in L0L0

It is shown that every centralizer from any “metric function space” XX to L0L0 is continuous at the origin of XX. As a consequence, every short exact sequence of L∞L∞-modules 0→L0→Z→X→00→L0→Z→X→0 splits if XX is a “minimal” function space, and in particular if X=L0X=L0. There are pairs of Orlicz function spaces U,VU,V such that Hom(U,V)=0, but Ext(U,V)≠0.