On injective tensor powers of 1

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Abstract

In this paper we prove that the 3-fold injective tensor product 1ˆε1ˆε1 is not isomorphic to any subspace of 1ˆε1. This result provides a new solution to a problem of Diestel on the projective tensor products of c0. Moreover, this result implies that for any infinite countable compact metric space K, the 3-fold projective tensor product C(K)ˆπC(K)ˆπC(K) is not isomorphic to any quotient of C(K)ˆπC(K).

Introduction

For standard Banach space terminology employed throughout the paper the reader is referred to [6] and [8]. For nN, a tensor norm α, and a Banach space X, let ˆαnX denote the n-fold α-tensor product of X with itself.

Very recently the authors solve a problem attributed to Diestel [3, Theorem 1.3] by proving that ˆπ3c0 is not isomorphic to ˆπ2c0. In the present paper we consider two natural problems that arise from this result. The first problem is whether this result extends to C(K) spaces other than c0, here the space C(K) will stand for the Banach space of all continuous, real-valued functions on the compact Hausdorff space K and equipped with the supremum norm. The first problem can be precisely stated as:

Problem 1.1

Let K be an infinite compact Hausdorff space. Is it true that ˆπ3C(K) is not isomorphic to ˆπ2C(K)?

The second problem is to know if the dual spaces of ˆπ3c0 and ˆπ2c0 are isomorphic to each other. By using well-known properties of projective and injective tensor products [8] this problem can be rewritten as follows:

Problem 1.2

Is ˆε31 isomorphic to ˆε21?

This last problem was proposed to us by Richard M. Aron to whom we are grateful for the interest shown in this research topic.

The main goal of this paper is to present a negative solution to Problem 1.2. This follows directly from the following theorem.

Theorem 1.3

ˆε31 is not isomorphic to any subspace of ˆε21.

Regarding Problem 1.1, first note that if K is an infinite countable compact metric space, then (C(K)ˆπC(K)ˆπC(K)) is linearly isometric to 1ˆπ1ˆπ1. Indeed, since C(K)ˆπC(K) is c0-saturated [5], it is easy to see that every bounded linear operator from C(K)ˆπC(K) to 1 is compact. In other words, the space L(C(K)ˆπC(K),1) is equal to the space K(C(K)ˆπC(K),1). On the other hand, keeping in mind that C(K) is isomorphic to 1 [6, p.20], (C(K)ˆπC(K)ˆπC(K)) identifies itself linearly and isometrically to L(C(K)ˆπC(K),1). Thus, we conclude that (C(K)ˆπC(K)ˆπC(K)) is linearly isometric to (C(K)ˆπC(K))ˆε1, which in turn is linearly isometric to 1ˆπ1ˆπ1.

Therefore it follows from Theorem 1.3 that ˆπ3C(K) is not isomorphic to any quotient of ˆπ2C(K). In particular, Problem 1.1 has a positive solution when K is an infinite countable compact metric space.

Of course it would be interesting to know if Problem 1.1 also has a positive solution when K is the interval of real numbers [0,1] or K is βN, the Stone-Cech compactification of the discrete set of natural numbers N, see [2] to some geometric properties of the spaces C([0,1])ˆπC([0,1]) and C(βN)ˆπC(βN).

Theorem 1.3 also provides a new proof that ˆε21 is not isomorphic to any subspace of 1 [7, Corollary 2.1]. However we do not know how to solve:

Problem 1.4

Suppose that for some m,nN with m,n3, ˆεm1 is isomorphic to ˆεn1. Is it true that m=n?

Before presenting the proof of Theorem 1.3, we find it interesting to highlight here the technical difference that exists between the solutions of Diestel's problem obtained in [3] and that given by Theorem 1.3. The fundamental property used in [3] concerned 2 upper estimates on the branches of weakly null trees in the 2-fold tensor product ˆπ2c0. Trees dualize nicely, but the dual property to upper 2 estimates on weakly null trees in some Banach space X is lower 2 estimates on the branches of weak null trees in X. Therefore the result from [3] does not yield that there is no isomorphic embedding of ˆε31 into ˆε21, because such an isomorphic embedding need not be weak-weak continuous. Thus Theorem 1.3 is not a trivial consequence of the result of [3].

Also, the results of [3] were stated in terms of weakly null trees, but the objects produced were weakly null arrays, which can be viewed as a special kind of weakly null tree. Since weakly null arrays are weakly null trees, 2 upper estimates on the branches of weakly null trees imply the same estimates on the branches of weakly null arrays, but the converse need not hold [1]. Therefore the existence of weakly null arrays which do not satisfy a uniform 2 upper estimate is a stronger condition than the existence of weakly null trees. In the current work, we use the fact that [3] produced arrays and not simply sequences, as this allows us to circumvent the difficulty that isomorphic embeddings need not be weak-weak continuous. The key step is noting that arrays are amendable to a certain differencing procedure, while the same differencing procedure cannot be applied to trees. This differencing is used here to overcome a difficulty not present in [3].

Section snippets

Proof of Theorem 1.3

For a Banach space X and nN, a family (xik)i=1,k=1,n of X is called an n-array. For C>0, an n-array is said to be C-separated provided that for any 1kn and any distinct i,jN, xikxjkC.

For a Banach space X and nN, let δn(X) denote the infimum of d>0 such that for any C>0 and any bounded, C-separated n-array (xik)i=1,k=1,n in X, there exist i1<j1<<in<jn such thatdk=1n(xikkxjkk)Cn1/2.

Obviously if X is isomorphic to a subspace of Y, then supnδn(X)/δn(Y)<. More precisely, if X,Z

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