Solution to a problem of Diestel

Abstract

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On injective tensor powers of ℓ1

“…Note that there exists a constant 0 < β (independent of both n and k) such that β log(n) t n k ℓ1 . It was shown in [3] that there exists a constant τ < ∞ (independent of n) such that…”

“…, 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].…”

“…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].…”

“…For n ∈ N, a tensor norm α, and a Banach space X, let ⊗ n α X 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 ⊗ 3 π c 0 is not isomorphic to ⊗ 2 π c 0 . In the present paper we consider two natural problems that arise from this result.…”

“…The fundamental property used in [3] concerned ℓ 2 upper estimates on the branches of weakly null trees in the 2-fold tensor product ⊗ 2 π c 0 . Trees dualize nicely, but the dual property to upper ℓ 2 estimates on weakly null trees in some Banach space X is lower ℓ 2 estimates on the branches of weak * null trees in X * .…”