We prove that the Kalton centralizer on L p [0, 1], for 0 < p < ∞, is not strictly singular: in all cases there is a Hilbert subspace on which it is trivial. Moreover, for 0 < p < 2 there are copies of q , with p < q < 2, on which it becomes trivial. This is in contrast to the situation for p spaces, in which the Kalton-Peck centralizer is strictly singular.
We provide three new examples of twisted Hilbert spaces by considering properties that are "close" to Hilbert. We denote them Z(J ), Z(S 2 ) and Z(T 2 s ). The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, Z(S 2 ) and Z(T 2 s ) are not asymptotically Hilbertian. Moreover, the space Z(T 2 s ) is a HAPpy space and the technique to prove it gives a "twisted" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987-2001, 2012. This is, we can construct a nontrivial twisted Hilbert space such that the isomorphism constant from its n-dimensional subspaces to ℓ n 2 grows to infinity as slowly as we wish when n → ∞.
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