Singular twisted sums generated by complex interpolation

Abstract

This paper deals with extensions or twisted sums of Banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension. The results are presented in the context of interpolation of families of Banach spaces, and are applied to the study of submodules of Schatten classes. We also obtain nontrivial extensions of spaces without the CAP which also fail the CAP.

Derivation of vector-valued complex interpolation scales

“…Reasoning as in [21, p.1165] (see also [11,Section 3.3]) one gets that if a 0 (x), a 1 (x) is an almost optimal Lozanovskii decomposition for x then…”

“…• According to [11,Prop. 3.6], λ p = (λ, ℓ ∞ ) 1/p with associated derivation p K. Therefore, by the reiteration theorem [1, Section 4.6], one has [11,Prop. 3.7].…”

“…The paper [11] studied the nontriviality and singularity of the sequences Ω θ in terms of the initial couple (X 0 , X 1 ). More precisely, let X be a Banach space with a 1-unconditional basis.…”

“…More precisely, let X be a Banach space with a 1-unconditional basis. Following [11], we consider the parameter A X (n) = sup{ x 1 + . .…”

“…While it is relatively easy to get (X, X * ) 1/2 = ℓ 2 , it is rather difficult to calculate the associated derivation and study its properties. The paper [3] presents a complete description of the derivations that appear when considering scales of Lorentz spaces, while the paper [11] performs a thorough study of singular derivations. In this paper we continue the previous work by obtaining new examples of derivations, study their properties and show that the hypotheses in the main theorems of [11] are actually not necessary.…”