The Kalton centralizer on \(L_p[0,1]\) is not strictly singular (Q2845420)

The title refers to a naturally defined quasilinear mapping on \(L_p(0,1)\), where \(0<p<\infty\). The restriction of this map to the span \(R\) of the Rademacher functions is shown to be trivial, i.e., \(L_p(0,1)\) is complemented in the associated twisted sum of itself with \(R\). Of course, \(R\) is isomorphic to \(\ell_2\); it is shown furthermore that, if \(p<q<2\), then the Kalton centralizer is also trivial when restricted to a certain subspace isomorphic to \(\ell_q\). These results contrast with the situation on \(\ell_p\), where the corresponding centralizer, restricted to any infinite-dimensional subspace, remains nontrivial.