The Kalton centralizer on 𝐿_{𝑝}[0,1] is not strictly singular

Abstract

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.

On twisted sums of Schreier spaces

“…Moreover, it was proved in [3] that the Kalton-Peck map is strictly singular on a number of spaces including Tsirelson space and L p for 2 ≤ p < ∞ regarded as a Banach space with unconditional basis through the Haar system. As it was mentioned in the introduction, replacing the Kalton-Peck map by the natural Kalton centralizer on L p , it was proved in [12] that the Kalton map is not strictly singular on L p for every 0 < p < ∞, see also [1]. The reader should also check the recent paper [8] where a connection between singularity and interpolation is established.…”

“…In this latter case, the singularity fails. The author has studied in [12] the singularity for the Schatten classes S p giving a criterion for the corresponding B(H)-submodules of S p . To finish, the paper [3] studies the singularity for the Kalton-Peck map.…”

Type, cotype and twisted sums induced by complex interpolation

“…For 1 < p < ∞ these are nontrivial twisted sums, but unlike the previous case these are not singular ( [39], see also [5]). Actually, the extension is trivial on the copy of ℓ 2 spanned by the Rademacher functions, and if 1 < p < q < 2, then it is trivial on a copy of ℓ q .…”

Singular twisted sums generated by complex interpolation

“…In [6] it was shown that no L ∞ -centralizer on L p is singular for 0 < p < ∞; previously, it had been shown in [38] that the Kalton-Peck L ∞ -centralizer Ω(f ) = f log |f |/ f on L p is not singular since it becomes trivial on the Rademacher copy of ℓ 2 . Proposition 5.3 tells us that it is not trivial on any subspace generated by disjointly supported vectors.…”