On strictly singular nonlinear centralizers

We show that c(0) is the only Banach space with unconditional basis that satisfies the equation Ext(X, X) = 0. This partially improves an old result by Kalton and Peck. We prove that the Kalton-Peck maps are strictly singular on a number of sequence spaces, including l(p) for 0 < p < infinity, Tsirelson and Schlumprecht spaces and their duals, as well as certain super-reflexive variations of these spaces. In the last section, we give estimates of the projection constants of certain finite-dimensional twisted sums of Kalton-Peck type. (C) 2011 Elsevier Ltd. All rights reserved.