Bounded approximation properties in terms of $C[0,1]$

Abstract

Let X be a Banach space and let I be the Banach operator ideal of integral operators. We prove that X has the λ-bounded approximation property (λ-BAP) if and only if for every operator T ∈ I (X, C[0, 1] * ) there exists a net (S α ) of finite-rank operators on X such that S α → I X pointwise and lim supWe also prove that replacing I by the ideal N of nuclear operators yields a condition which is equivalent to the weak λ-BAP.

Bounded approximation properties via Banach operator ideals

“…To prove this result (see [14]), one develops further ideas of the proof of Theorem 4.1 in [14] and also relies on the fact that the BAP is separably determined with respect to ideals as expressed below. [15] use, for instance, the fact that the Banach operator ideal I is injective with respect to norm-preserving extension operators (see [15,Proposition 2.1]) and the Grothendieck-Sakai characterization of L 1 -preduals (or Lindenstrauss's spaces). The proof of the non-separable case in Theorem 4.4 (see [15]) is rather involved.…”

“…[15] use, for instance, the fact that the Banach operator ideal I is injective with respect to norm-preserving extension operators (see [15,Proposition 2.1]) and the Grothendieck-Sakai characterization of L 1 -preduals (or Lindenstrauss's spaces). The proof of the non-separable case in Theorem 4.4 (see [15]) is rather involved. It is quite natural that the λ-BAP of X is obtained by showing that every separable ideal of X has the λ-BAP (see Proposition 4.1) and using the separable case of Theorem 4.4.…”

Absolutely summing operators on separable Lindenstrauss spaces as tree spaces and the bounded approximation property

“…Remark 3.5. Theorems 1.3 and 1.4 in [15] assert that the equivalences (a) ⇔ (b) of Theorems 3.3 and 3.4 hold in the particular case when Z = C[0, 1]. In the above characterizations of the λ-BAP and the weak λ-BAP, one may, e.g., take Z to be any separable Lindenstrauss space whose dual space is non-separable, in particular, one may take Z = M or Z = C(∆).…”

“…which is a contradiction. [15] reveals that they go through if C[0, 1] is replaced by any Banach space Z which is isometrically universal for all separable Banach spaces and such that Z * has the 1-BAP. By Proposition 3.2, Z is isometrically universal for all separable Banach spaces.…”

“…In [15], we characterized the λ-BAP and the weak λ-BAP in terms of the space of integral operators I(X, C[0, 1] * ) and the space of nuclear operators N (X, C[0, 1] * ), respectively. In Section 3, we show that C[0, 1] can be replaced by any Lindenstrauss space Z such that Z * fails to have the Radon-Nikodým property and we still obtain characterizations of the λ-BAP and the weak λ-BAP.…”