Approximation properties of tensor norms and operator ideals for Banach spaces

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.


Introduction
The main subjects of this paper originate from the classical approximation properties for Banach spaces, which was systematically investigated by Grothendieck [1]. A Banach space X is said to have the approximation property (AP) if where id X is the identity map on X, is the ideal of finite rank operators and τ c is the topology of uniform convergence on compact sets. Let X and Y be Banach spaces. We denote by ⊗ X Y the algebraic tensor product of X and Y. The normed space ⊗ X Y equipped with a norm α is denoted by ⊗ X Y α and its completion is denoted by ⊗ X Y α  . The basic two norms on ⊗ X Y are the injective norm ε and the projective norm π which are defined as follows. generated tensor norm α, a Banach space X is said to have the α-AP if for every Banach space Y, the natural map is injective (cf. [2,Section 21.7]). It is well known that if a Banach space X has the AP, then it has the α-AP for every finitely generated tensor norm α (cf. [2, Proposition 21.7(1)]). Some of the well-known tensor norms can be obtained from the tensor norm α p q , ( ≤ ≤∞ / + p q p 1 , ,1 / ≥ q 1 1), which was introduced by Lapresté [4]. For ≤ < ∞ p 1 , ℓ ( ) X p w stands for the Banach space of all Xvalued weakly p-summable sequences endowed with the norm ∥⋅∥ p w . Let ≤ ≤ ∞ , -AP in terms of certain approximation properties of operator ideals. As a consequence, it was shown that a Banach space X has the α p q , -AP if it has the α p,1 -AP.
It is well known that a Banach space X has the λ-bounded AP if and only if for every Banach space Y, the natural map ). More generally, for a finitely generated tensor norm α, a Banach space X is said to have the λ-bounded α-AP if for every Banach space Y, the natural map The main goal of this paper is to study the α-AP and the λ-bounded α-AP in terms of operator ideals. In Section 2, we extend the result of Díaz et al. [7], and in Section 3, we obtain some bounded versions of the results obtained in Section 2. As an application, it is shown that a Banach space X has the λ-bounded α p q , -AP if it has the λ-bounded α p,1 -AP. Consequently, if X has the λ-bounded α p,1 -AP, then X has the λ-bounded * α p p , -AP.
be the maximal Banach operator ideal associated with a finitely generated tensor norm α. Then for all Banach spaces X and Y, ( Let α be a finitely generated tensor norm. According to [2, Proposition 21.7(4)], a Banach space X has the α-AP if and only if for every Banach space Y, the natural map be the maximal Banach operator ideal associated with a finitely generated tensor norm α. Then the following statements are equivalent for a Banach space X. (a) X has the α-AP.
with the weak * topology on ( ) where〈⋅ ⋅〉 , is the dual action on ( ) as → ∞ n , and for every n, . The aforementioned argument also shows that is the natural map. As in the proof of (a) ⇒ (b), we see that for every ∈ * * x X and ∈ * * y Y , and so with the weak * topology on ( ) with the weak * topology on ( ) Recall that a Banach space X has the AP if and only if X has the π-AP. Then the most special case of Corollary 2.3 is the following.
Corollary 2.4. The following statements are equivalent for a Banach space X (a) X has the AP.
Proof. It is well known that π is associated with the ideal of integral operators and = adj holds isometrically (cf. [2]). Since = π π t , we have the conclusion. □ be the maximal Banach operator ideal associated with a finitely generated tensor norm α. Then a Banach space X has the α t -AP if and only if for every Banach space Y, ( Proof. Assume that X has the α t -AP. Let Y be a Banach space. By Lemma 2.1, Then as in the proof of Theorem 2.2, we can show that the natural map is not injective. This contradicts the assumption that X has the α t -AP.
To show the converse, let Y be a Banach space. We want to show that the natural map

The bounded α-approximation property
Let α be a tensor norm and let X and Y be Banach spaces. Recall from [2, 12.4] that for every ∈ ⊗ u X Y, let Proposition 21.7(2)], a Banach space X has the λ-bounded α-AP if and only if for every Banach space Y, ; , it follows that a Banach space X has the λ-bounded α t -AP if and only if for every Banach space Y, for every ∈ ⊗ u X Y.
Thus, for every ∈ x X and ∈ y Y , , with / + / ≥ p q 1 1 1 and let ≥ λ 1. If a Banach space X has the λ-bounded g p -AP, then X has the weak λ-BAP for * * p q , .
Proof. By Corollary 3.5 and Lemma 3.6, if X has the λ-bounded g p -AP, then for every Banach space Z and every ∈ ( )