BOUNDED APPROXIMATION PROPERTIES IN TERMS OF C [0 , 1]

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BOUNDED APPROXIMATION PROPERTIES IN TERMS OF C [0 , 1]

ÅSVALD LIMA, VEGARD LIMA and EVE OJA^∗

Abstract

LetXbe a Banach space and letIbe the Banach operator ideal of integral operators. We prove thatXhas theλ-bounded approximation property (λ-BAP) if and only if for every operator T ∈I(X, C[0,1]^∗)there exists a net(Sα)of finite-rank operators onXsuch thatSα →IX

pointwise and

lim sup

α TS_αI ≤λTI.

We also prove that replacingI by the idealN of nuclear operators yields a condition which is equivalent to the weakλ-BAP.

1. Introduction

LetXandY be Banach spaces. We denote byL(X, Y )the Banach space of all bounded linear operators fromXtoY, and we writeL(X)forL(X, X). The subspace ofL(X)of finite-rank operators is denoted byF(X). LetIX

denote the identity operator onX.

Recall that a Banach spaceX is said to have theapproximation property (AP) if there exists a net(Sα) ⊂ F(X) such thatSα → IX uniformly on compact subsets ofX. If(Sα)can be chosen with sup_αSα ≤ λfor some λ≥1, thenXis said to have theλ-bounded approximation property(λ-BAP).

LetA =(A, A)be a Banach operator ideal. Recently, an approximation property which is bounded forAwas introduced and studied in [11] as follows.

We say thatXhas theλ-bounded approximation property forA (λ-BAP for A) if for every Banach spaceY and every operatorT ∈A(X, Y )there exists a net(Sα) ⊂ F(X)such thatSα → IX uniformly on compact subsets ofX

and lim sup

α TS_αA ≤λTA.

The λ-BAP for A extends the notion of the weak λ-BAP which is, by definition, theλ-BAP for the idealW of weakly compact operators. The weak

∗The research of Eve Oja was partially supported by Estonian Science Foundation Grant 7308 and Estonian Targeted Financing Project SF0180039s08.

Received 14 July 2010, in final form 29 October 2010.

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BAP was introduced in [12] and studied in [11], [12], [13], [17], [18], [19], [20]. It is immediate that the λ-BAP implies the λ-BAP for every Banach operator idealA (since TS_αA ≤ TASα), and it is equivalent to the λ-BAP for the idealL of all bounded linear operators.

By [17] (see [20] for a simpler proof), the weakλ-BAP and theλ-BAP are equivalent for a Banach spaceXwheneverX^∗orX^∗∗has the Radon–Nikodým property. It remains open whether the weakλ-BAP is strictly weaker than the λ-BAP. If they were equivalent, then, by [12], the answer to the long-standing famous open problem (Problem 3.8 in [1]), whether the AP of a dual Banach space implies the 1-BAP, would be “yes”. For a recent survey on bounded approximation properties, see [21].

In [11], it was proved that the BAP is precisely the BAP for the idealI of integral operators, and the weak BAP is precisely the BAP for the idealN of nuclear operators. In [11], it was also proved that in these cases the requirement

“for every Banach spaceY” can be relaxed by takingY =^∗_∞for the BAP and Y =c0^∗for the weak BAP. More precisely, the following holds.

Theorem 1.1 (see [11, Theorem 2.1 and Proposition 4.2]). Let X be a Banach space, and let1≤λ <∞. The following statements are equivalent.

(a) Xhas theλ-BAP.

(b) For every Banach spaceY and every operatorT ∈I(X, Y )there exists a net(Sα) ⊂ F(X)such thatSα → IX uniformly on compact subsets ofXand

lim sup

α TS_αI ≤λTI.

(c) For every T ∈ I(X, ^∗_∞) there exists a net (Sα) ⊂ F(X)such that Sα →IXpointwise and

lim sup

α TS_αI ≤λTI.

Theorem 1.2 (see [11, Theorem 3.1 and Proposition 4.1]). Let X be a Banach space, and let1≤λ <∞. The following statements are equivalent.

(a) Xhas the weakλ-BAP.

(b) For every Banach spaceYand every operatorT ∈N(X, Y )there exists a net(Sα) ⊂ F(X)such thatSα → IX uniformly on compact subsets ofXand

lim sup

α TS_αN ≤λTN.

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(c) For every T ∈ N(X, c^∗0) there exists a net (Sα) ⊂ F(X)such that Sα →IXpointwise and

lim sup

α TS_αN ≤λTN.

The classical spacesc0and∞are, indeed, very different from each other.

A natural question would be: can the spaces c0 and∞ be replaced by one classical Banach space, preferably separable, which would characterize both the BAP and the weak BAP? Our main aim of this paper is to show that the spaceC[0,1] of continuous functions fits for the both BAPs. Our main results are as follows (conditions (b) below are to be compared with conditions (c) of Theorems 1.1 and 1.2).

Theorem1.3.LetXbe a Banach space, and let1≤λ <∞. The following statements are equivalent.

(b) For everyT ∈I(X, C[0,1]^∗)there exists a net(Sα)⊂F(X)such that Sα →IXpointwise and

lim sup

α TS_αI ≤λTI.

Theorem1.4.LetXbe a Banach space, and let1≤λ <∞. The following statements are equivalent.

(a) Xhas the weakλ-BAP.

(b) For every T ∈ N(X, C[0,1]^∗)there exists a net(Sα) ⊂ F(X)such thatSα →IXpointwise and

lim sup

α TS_αN ≤λTN.

Theorem 1.4 and the separable case of Theorem 1.3 will be proved in Sec- tion 2 relying on the fact that the Banach operator idealI is injective with respect to norm-preserving extension operators (see Proposition 2.1). The non- separable case of Theorem 1.3 will be deduced from the separable case in Section 3 relying on the main result of Section 3 (Theorem 3.2) stating that a property ofX, similar to conditions (c) of Theorems 1.1 and 1.2 and to conditions (b) of Theorems 1.3 and 1.4, is inherited by ideals in Banach spaces.

Our notation is standard. A Banach spaceXwill be regarded as a subspace of its bidualX^∗∗under the canonical embeddingjX:X →X^∗∗. The closure of a setA⊂ Xis denotedA. The tensor productX⊗Y with a tensor norm

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α is denoted byX⊗α Y and its completion byX ˆ⊗α Y. We shall use only the classical projective tensor normπ = π and the injective tensor norm ε. SinceF(X, Y )=X^∗⊗Y, we shall writeTπ forT ∈F(X, Y )( _πis called the finite nuclear norm in [22]). Let us recall that, for Banach operator ideals A and B, the inclusion A ⊂ B means that A(X, Y ) ⊂ B(X, Y ) and TA ≥ TB for all Banach spaces X and Y and for all operators T ∈A(X, Y ).

We refer to the books by Diestel and Uhl [3] and Ryan [23] for the classical approximation properties, tensor products, and for the common Banach operator ideals such asN andI; see also [2] by Diestel, Jarchow, and Tonge and Pietsch’s book [22] for operator ideals. We use “Banach operator ideal”

for “normed operator ideal” in [22], or for “Banach ideal” in [2] and [23] (note that, in the Banach spaces context, the term “ideal” has its own meaning (see Section 2)).

2. Proofs of Theorem 1.4 and the separable case of Theorem 1.3 Recall that a Banach operator idealAisinjectiveifJ TA = TAwhenever T ∈ A(X, Y ) andJ ∈ L(Y, Z)is an into isometry. It is well known that the Banach operator idealI of integral operators is not injective (see, e.g., [22, 8.4.10]). Our first result shows thatI is injective with respect to norm- preserving extension operators, a fact which will be used in the proofs of Theorems 1.4 and 2.6 below.

LetYbe a closed subspace of a Banach spaceZ. An operator∈L(Y^∗,Z^∗) is called anextension operatorif(y^∗)(y)= y^∗(y)for ally^∗ ∈ Y^∗and all y ∈ Y. IfY admits an extension operator ∈ L(Y^∗, Z^∗), which is norm- preserving (i.e., =1), thenY is called anidealinZ. This is equivalent to the annihilatorY^⊥ofY being the kernel of a norm one projection inZ^∗.

Proposition2.1.Let Xbe a Banach space. Let Y be a closed subspace of a Banach space Z. If there exists a norm-preserving extension operator ∈L(Y^∗, Z^∗), thenTI = TI wheneverT ∈I(X, Y^∗).

Proof. We are going to use well-known facts about tensor products (see, e.g., [3] or [23]). SinceI(X, Y^∗)=(X⊗εY )^∗andI(X, Z^∗)=(X⊗εZ)^∗, we may considerT ∈(X⊗εY )^∗andT ∈(X⊗εZ)^∗. Taking into account thatX⊗εYis a subspace ofX⊗εZ, let us observe thatT extendsT. Indeed, for allx ∈Xandy∈Y,

(T )(x⊗y)=(T x)(y)=(T x)(y)=T (x⊗y).

Hence,TI ≥ TI. On the other hand,TI ≤ TI = TI.

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Recall that a Banach space is aPλ-space, for someλ≥1, if it is complemented, by a projection whose norm does not exceedλ, in any Banach space containing it (as an isometrically isomorphic subspace). The next result is due to Fakhoury [4, Corollary 3.3]. Fakhoury’s proof relies on Lindenstrauss’s Memoir [14] and his own results established in [4]. For a simple direct proof, see [16, Proposition 5.3].

Proposition2.2.LetY be a closed subspace of a Banach spaceZ. IfY^∗∗

is aPλ-space, then there exists an extension operator∈ L(Y^∗, Z^∗)with ≤λ.

It is well known that, for every set, the space∞()is aP1-space (see, e.g., [15, p. 105]). In particular,c0^∗∗=∞ is aP1-space. More generally,Y^∗∗

is aP1-space wheneverYis anL¹-predual, i.e.,Y^∗is isometrically isomorphic to a space of typeL¹( , μ)(see, e.g., [26, p. 1706]).

Corollary2.3.Let Y be an L1-predual (in particular,Y = c0). IfY is contained in a Banach spaceZ (as an isometrically isomorphic subspace), thenY is an ideal inZ.

On the other hand, the following holds.

Proposition2.4 (see [4, Proposition 3.4]). Every ideal in anL¹-predual is anL¹-predual itself.

Proof. Since [4] considers only the real case and does not provide a proof, we include a proof for completeness. Thus, letY be an ideal in anL1-predual Z, and let ∈ L(Y^∗, Z^∗) be a norm-preserving extension operator. Since Z^∗∗is aP1-space and^∗ provides a norm one projection inZ^∗∗ ontoY^∗∗, Y^∗∗is also aP¹-space (it is easily seen that 1-complemented subspaces of a P1-space areP1-spaces). Hence, by the Grothendieck–Sakai theorem (see [5]

for the real case and [24] for the complex case),Y is anL1-predual.

Let us first prove Theorem 1.4.

Proof of Theorem 1.4. By Theorem 1.2, we only need to prove the implication (b)⇒(a). For this, it suffices to show that condition (b) of The- orem 1.4 implies condition (c) of Theorem 1.2.

Let T ∈ N(X, c^∗0). Since c0 embeds isometrically inC[0,1], by Corol- lary 2.3 there exists a norm-preserving extension operator∈L(c^∗0,C[0,1]^∗). SinceT ∈N(X, C[0,1]^∗), there exists(Sα) ⊂ F(X)such thatSα →IX

pointwise and lim sup

α TS_αN ≤λTN ≤λTN =λTN.

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It is well known (see, e.g., [23, p. 176]) that for a finite-rank operator, acting to a space with the metric AP, its nuclear and integral norms coincide. Hence, TS_αN = TS_αI andTS_αN = TS_αI. Using Proposition 2.1, we therefore have

TS_αN = TS_αI = TS_αI = TS_αN. Hence,

lim sup

α TS_αN ≤λTN

as desired.

Remark2.5. It is an easy exercise to show thatc0^∗=1embeds isometrically inC[0,1]^∗. It seems that an arbitrary into isometry∈L(c^∗0, C[0,1]^∗) cannot be used for proving Theorem 1.4.

Theseparable caseof Theorem 1.3 is immediate from Theorem 2.6 below and Theorem 1.1.

Theorem2.6.LetX be a separable Banach space, and let1 ≤ λ < ∞. If for everyT ∈ I(X, C[0,1]^∗) there exists a net(Sα) ⊂ F(X)such that Sα →IXuniformly on compact subsets ofX(respectively, pointwise)and

lim sup

α TS_αI ≤λTI,

then for everyT ∈I(X, ^∗_∞)there exists a net(Sα) ⊂ F(X)with the same properties.

Proof. LetT ∈I(X, ^∗_∞). Since ranTis separable, by a result of Sims and Yost [25] (see [6, p. 138]), we can find a separable idealY in∞which admits a norm-preserving extension operator ∈ L(Y^∗, ^∗_∞) satisfying ranT ⊂ ran. By Proposition 2.4,Y is anL¹-predual.

Letj :Y →∞ denote the identity embedding. Observe that T =j^∗T .

Indeed, letx∈X. Since ranT ⊂ran, there isy^∗∈Y^∗such thatT x=y^∗. Hence,j^∗T x=j^∗y^∗=IY^∗y^∗=y^∗=T x.

SinceY is separable, it embeds isometrically inC[0,1]. By Corollary 2.3, there exists a norm-preserving extension operator ∈L(Y^∗, C[0,1]^∗). Since j^∗T ∈I(X, C[0,1]^∗), there exists a net(Sα)⊂F(X)such thatSα →IX

and

lim sup

α j^∗TS_αI ≤λj^∗TI ≤λTI.

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On the other hand, using Proposition 2.1 twice, we have

TS_αI = j^∗TS_αI = j^∗TS_αI = j^∗TS_αI. From this, the desired inequality is immediate.

Remark2.7. In the above proof of Theorem 2.6, we applied Proposition 2.4 to show that an idealY in ∞ is an L1-predual. An alternative proof of this fact, relying on intersection properties of balls, can be done as follows. By results of Lindenstrauss [14] (the real case) and Hustad [8] (the complex case) (see [9, Theorem 4.1] and [10, Theorem 5.8]),Y is anL1-predual if and only ifY is an almostE(n)-space for alln∈^N. Recall (see [8] and [10, p. 9]) that a Banach spaceY is analmostE(n)spaceif for each family ofnclosed balls B(y¹, r¹), . . . , B(yn, rn)inY the following implication holds:

n

i=1

B(y^∗(yi), ri)=∅ ∀y^∗∈Y^∗, y^∗ ≤1

⇒

n

i=1

B(yi, ri +ε)=∅ ∀ε >0.

Lete_k^∗∈^∗_∞be the coordinate functionals, and lety_k^∗∈Y^∗be their restrictions to Y. If the above assumption holds, then there exist numbers ak such that

|e_k^∗(yi)−ak| = |y_k^∗(yi)−ak| ≤rifor alli =1, . . . , n. Hence,x :=(ak)∈∞

andyi−x ≤riin∞for alli =1, . . . , n. But thenyi−^∗x = ^∗(yi− x) ≤ ri inY^∗∗for alli = 1, . . . , n. This implies, by the principle of local reflexivity, that for everyε >0 there existsyε ∈Ysuch thatyi−yε ≤ri+ε for alli =1, . . . , n, as desired.

3. Proof of the non-separable case of Theorem 1.3

The proof of the non-separable case of Theorem 1.3 relies on the following reformulation of the BAP in terms of separable ideals.

Theorem 3.1 (see [11, Proposition 4.3 and Theorem 2.2]). Let X be a Banach space, and let1≤λ <∞. The following statements are equivalent.

(b) Every separable idealZinXhas theλ-BAP.

The next result is the main theorem of this section. Its assumptionA ⊂W can be equivalently expressed as follows: ifT ∈A(X, Y ), then ranT^∗∗⊂Y. This assumption holds for many operator ideals. For us, it is important that I ⊂W.

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Theorem3.2.LetXandYbe Banach spaces, letAbe a Banach operator ideal such thatA ⊂ W, and let1 ≤ λ < ∞. Assume thatX has the weak BAP. IfX has the property that for every T ∈ A(Y, X^∗) there exists a net (Sα)⊂F(X)such thatSα →IXpointwise and

lim sup

α S^∗_αTA ≤λTA, then every idealZinXhas the same property.

Proof. LetT ∈A(Y, Z^∗). We consider the set of allν=(ε, K, L), where ε >0, andK ⊂ZandL⊂Z^∗are finite sets. We need to prove that for every ν=(ε, K, L)there existsUν ∈F(Z)such that

|z^∗(Uνz−z)|< ε ∀z∈K, ∀z^∗∈L,

and U_ν^∗TA ≤λTA +ε.

Indeed, this would imply thatUν →IZin the weak operator topology and lim sup

ν U_ν^∗TA ≤λTA.

Hence, passing to a net of convex combinations far out in(Uν), we could assume thatUν →IZin the strong operator topology, as desired.

Let us fixν=(ε, K, L). Let∈L(Z^∗, X^∗)be a norm-preserving extension operator. ThenT ∈A(Y, X^∗), and there existsS= Sα ∈F(X)such

that Sz−z< ε

2 max{z^∗:z^∗∈L} ∀z∈K,

and S^∗TA ≤λTA + ε

2 ≤λTA+ ε 2.

SinceXhas the weak BAP, there exists an extension operator∈X⊗X^∗^w

∗

⊂ L(X^∗, X^∗∗∗) = (X^∗ ˆ⊗π X^∗∗)^∗ (see [13, Propositions 2.1, 2.3, and 2.5]

and [20, Corollary 3.18]). Then ∈ L(Z^∗, X^∗∗∗) = (Z^∗ˆ⊗πX^∗∗)^∗. We show that ∈ Z⊗X^∗^w

∗

. Letu = _∞

n=1z^∗_n ⊗x_n^∗∗ ∈ Z^∗ˆ⊗πX^∗∗, with

_∞

n=1z^∗_nx_n^∗∗<∞, and assume that u, z⊗x^∗ =

∞

n=1

z^∗_n(z)x_n^∗∗(x^∗)=0 ∀z∈Z, ∀x^∗∈X^∗.

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This means that_∞

n=1x_n^∗∗(x^∗)z_n^∗=0 inZ^∗for allx^∗∈X^∗, and therefore

∞

n=1

x_n^∗∗(x^∗)z^∗_n=0 ∀x^∗∈X^∗

inX^∗. Hence, denotingv=_∞

n=1z^∗_n⊗x_n^∗∗∈X^∗ˆ⊗πX^∗∗, we have u, =

∞

n=1

(z_n^∗)(x_n^∗∗)= v, =0,

because

v, x⊗x^∗ =

∞

n=1

(z_n^∗)(x)x_n^∗∗(x^∗)

=

∞

n=1

x_n^∗∗(x^∗)z^∗_n

(x)=0 ∀x∈X, ∀x^∗∈X^∗.

Since ∈ Z⊗X^∗^w^∗, there exists a net (Vβ) ⊂ F(X, Z)such that V_β^∗ → weak* inL(Z^∗, X^∗∗∗) = (Z^∗ˆ⊗πX^∗∗)^∗. We shall show that the desired operatorUν can be found in the formUν =VSiZ, whereiZ:Z→X denotes the identity embedding andV is a convex combination of operators Vβ.

SetH = ran(i_Z^∗S^∗). Then dimH <∞. LetiH : H →Z^∗be the identity embedding. Denote bySˆ the operatori_Z^∗S^∗considered as an operator to H. Then

iHSˆ =i_Z^∗S^∗,

and the operatorsSVˆ _β^∗T andSˆ^∗∗T belong toF(Y, H )= Y^∗⊗H. Since (Y^∗ˆ⊗πH )^∗ = L(Y^∗, H^∗) = F(Y^∗, H^∗) = Y^∗∗⊗H^∗ and we have (using that ranT^∗∗⊂Z^∗) that for ally^∗∗∈Y^∗∗andh^∗∈H^∗

y^∗∗⊗h^∗,SVˆ _β^∗T =h^∗(Sˆ^∗∗V_β^∗∗∗T^∗∗y^∗∗)=h^∗(SVˆ _β^∗T^∗∗y^∗∗)

=(Sˆ^∗h^∗)(V_β^∗T^∗∗y^∗∗)→

β T^∗∗y^∗∗⊗ ˆS^∗h^∗, = y^∗∗⊗h^∗,Sˆ^∗∗T, the net(SVˆ _β^∗T )β converges toSˆ^∗∗T weakly inY^∗ˆ⊗πH. Passing to a net of convex combinations far out in(Vβ), we may assume that our net(Vβ)also satisfies

ˆSV_β^∗T − ˆS^∗∗Tπ →

β 0,

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hence also

iHSVˆ _β^∗T −iHSˆ^∗∗TA →

β 0. Consequently,

(VβSiZ)^∗TA = iHSVˆ _β^∗TA →

β iHSˆ^∗∗TA.

A straightforward calculation shows thatiHSˆ^∗∗=i_Z^∗S^∗j_X^∗. Sincej_X^∗=IX^∗, iHSˆ^∗∗TA = i_Z^∗S^∗j_X^∗TA = i_Z^∗S^∗TA

≤ S^∗TA ≤λTA + ε 2. Hence, there is someβ0such that forβ ≥β0, one has

(VβSiZ)^∗TA ≤λTA +ε.

Finally, let us consider the operatorsSVˆ _β^∗,Sˆ^∗∗∈F(Z^∗, H ). Since for allz^∗∈Z^∗andh^∗∈H^∗

h^∗(SVˆ _β^∗z^∗)=(Sˆ^∗h^∗)(V_β^∗z^∗)→

β z^∗⊗ ˆS^∗h^∗, =h^∗(Sˆ^∗∗z^∗), SVˆ _β^∗ →β Sˆ^∗∗in the weak operator topology. Passing to a net of convex combinations far out in (SVˆ _β^∗), we may assume that SVˆ _β^∗z^∗ →β Sˆ^∗∗z^∗ for allz^∗ ∈ Z^∗. Hence alsoiHSVˆ _β^∗z^∗ →β iHSˆ^∗∗z^∗for allz^∗ ∈ Z^∗. This means, by calculations made above, that

(VβSiZ)^∗z^∗→

β i_Z^∗S^∗z^∗ ∀z^∗∈Z^∗. Letβ1be such that forβ≥β1, one has

|z^∗(VβSiZz)−(^∗Sz)(z^∗)|< ε

2 ∀z∈K, ∀z^∗∈L.

Since

^∗Sz−^∗z ≤ Sz−z< ε

2 max{z^∗:z^∗∈L} ∀z∈K, and(^∗z)(z^∗)=(z^∗)(z)=z^∗(z),

|(^∗Sz)(z^∗)−z^∗(z)|< ε

2 ∀z∈K, ∀z^∗∈L.

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Consequently, forβ ≥β1, one has

|z^∗(VβSiZz−z)|< ε ∀z∈K, ∀z^∗∈L.

SettingUν =VβSiZfor someβ ≥β⁰, β≥β¹, completes the proof.

To apply Theorem 3.2 in our context, we shall need the following result.

For a Banach operator idealA, let us denote byA^∗thedual operator ideal ofA. Its components areA^∗(X, Y )= {T ∈ L(X, Y ) : T^∗ ∈A(Y^∗, X^∗)}

withTA^∗ = T^∗A. (The notationA^∗means adjoint ideal in [2] and [22], where the dual operator ideal is denoted byA^d andA^dual, respectively.)

Proposition 3.3. Let X and Y be Banach spaces, let A be a Banach operator ideal such thatA ⊂ A^∗∗, and let1≤λ <∞. Letτ be a topology onL(X). The following statements are equivalent.

(a) For every T ∈ A(Y, X^∗)there exists a net (Sα) ⊂ F(X)such that Sα →IXinτand

lim sup

α S_α^∗TA ≤λTA.

(b) For everyT ∈ A^∗(X, Y^∗)there exists a net(Sα) ⊂ F(X)such that Sα →IXinτand

lim sup

α TS_αA^∗ ≤λTA^∗.

Proof. Below, we shall use the following observation. If X and Y are Banach spaces andT ∈L(X, Y^∗), then

T =j_Y^∗T^∗∗jX. Indeed,T =IY^∗T =j_Y^∗jY^∗T =j_Y^∗T^∗∗jX.

(a)⇒(b). ConsiderT ∈A^∗(X,Y^∗). ThenT^∗∈A(Y^∗∗,X^∗)andTA^∗ = T^∗A. SinceT^∗jY ∈A(Y, X^∗), there is(Sα)⊂ F(X)such thatSα →IX

inτ and

lim sup

α S_α^∗T^∗jYA ≤λT^∗jYA ≤λT^∗A =λTA^∗. On the other hand,

TS_αA^∗ = j_Y^∗T^∗∗S_α^∗∗jXA^∗ ≤ j_Y^∗T^∗∗S_α^∗∗A^∗

= S_α^∗T^∗jYA^∗∗ ≤ S_α^∗T^∗jYA. From this, the desired inequality is immediate.

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(b)⇒(a). Consider T ∈ A(Y, X^∗). Since A ⊂ A^∗∗, we have T ∈ A^∗∗(Y, X^∗) and thereforeT^∗ ∈ A^∗(X^∗∗, Y^∗). SinceT^∗jX ∈ A^∗(X, Y^∗), there is(Sα)⊂F(X)such thatSα →IXinτand

lim sup

α T^∗jXSαA^∗ ≤λT^∗jXA^∗ ≤λT^∗A^∗ =λTA^∗∗ ≤λTA. On the other hand,

S_α^∗TA = S_α^∗j_X^∗T^∗∗jYA ≤ S_α^∗j_X^∗T^∗∗A = T^∗jXSαA^∗. From this, the desired inequality is immediate.

SinceI =I^∗=I^∗∗, we have an immediate corollary, which we spell out for an easy reference.

Corollary3.4.LetXandY be Banach spaces, and let1≤λ <∞. Let τ be a topology onL(X). The following statements are equivalent.

(a) For every T ∈ I(Y, X^∗) there exists a net (Sα) ⊂ F(X)such that Sα →IXinτand

lim sup

α S_α^∗TI ≤λTI.

(b) For every T ∈ I(X, Y^∗)there exists a net (Sα) ⊂ F(X)such that Sα →IXinτand

lim sup

α TS_αI ≤λTI.

Proof of Theorem 1.3. By Theorem 1.1, we only need to prove the implication (b)⇒(a).

First of all, let us observe thatXhas the weakλ-BAP. Indeed, (b) of The- orem 1.3 implies (b) of Theorem 1.4, becauseN ⊂I and, as in the proof of Theorem 1.4,TS_αN = TS_αI wheneverTS_α ∈F(X, C[0,1]^∗). Accord- ing to Theorem 1.4,Xhas the weakλ-BAP.

By Corollary 3.4, (b)⇒(a), and Theorem 3.2, every separable idealZhas the property that for every T ∈ I(C[0,1], Z^∗) there exists a net (Sα) ⊂ F(Z) such that Sα → IZ pointwise and lim sup_αS_α^∗TI ≤ λTI. By Corollary 3.4, (a)⇒(b), and the separable version of Theorem 1.3, we get that every separable ideal Z inX has the λ-BAP. This means, according to Theorem 3.1, thatXhas theλ-BAP.

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