On injective tensor powers of ℓ1
Introduction
For standard Banach space terminology employed throughout the paper the reader is referred to [6] and [8]. For , a tensor norm α, and a Banach space X, let 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 is not isomorphic to . In the present paper we consider two natural problems that arise from this result. The first problem is whether this result extends to spaces other than , here the space 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 is not isomorphic to ? Problem 1.2 Is isomorphic to ?
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 is not isomorphic to any subspace of .
Regarding Problem 1.1, first note that if K is an infinite countable compact metric space, then is linearly isometric to . Indeed, since is -saturated [5], it is easy to see that every bounded linear operator from to is compact. In other words, the space is equal to the space . On the other hand, keeping in mind that is isomorphic to [6, p.20], identifies itself linearly and isometrically to . Thus, we conclude that is linearly isometric to , which in turn is linearly isometric to .
Therefore it follows from Theorem 1.3 that is not isomorphic to any quotient of . 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 or K is , the Stone-Cech compactification of the discrete set of natural numbers , see [2] to some geometric properties of the spaces and .
Theorem 1.3 also provides a new proof that is not isomorphic to any subspace of [7, Corollary 2.1]. However we do not know how to solve: Problem 1.4 Suppose that for some with , is isomorphic to . Is it true that ?
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 upper estimates on the branches of weakly null trees in the 2-fold tensor product . Trees dualize nicely, but the dual property to upper estimates on weakly null trees in some Banach space X is lower estimates on the branches of null trees in . Therefore the result from [3] does not yield that there is no isomorphic embedding of into , because such an isomorphic embedding need not be 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, 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 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 , a family of X is called an n-array. For , an n-array is said to be C-separated provided that for any and any distinct , .
For a Banach space X and , let denote the infimum of such that for any and any bounded, C-separated n-array in X, there exist such that
Obviously if X is isomorphic to a subspace of Y, then . More precisely, if
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