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2013 | 214 | 3 | 237-250
Tytuł artykułu

The classical subspaces of the projective tensor products of $ℓ_{p}$ and C(α) spaces, α < ω₁

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We completely determine the $ℓ_{q}$ and C(K) spaces which are isomorphic to a subspace of $ℓ_{p} ⊗̂_{π} C(α)$, the projective tensor product of the classical $ℓ_{p}$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from $ℓ_{p}$ to ℓ₁, 1 ≤ p < ∞.
The first main theorem is an extension of a result of E. Oja and states that the only $ℓ_{q}$ space which is isomorphic to a subspace of $ℓ_{p} ⊗̂_{π} C(α)$ with 1 ≤ p ≤ q < ∞ and ω ≤ α < ω₁ is $ℓ_{p}$. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pełczyński which allows us to classify, up to isomorphism, the separable spaces 𝓝(X,Y) of nuclear operators, where X and Y are direct sums of $ℓ_{p}$ and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K₁ and K₃ are finite or countable compact metric spaces of the same cardinality and 1 < p, q < ∞. Then, for any infinite compact metric spaces K₂ and K₄, the following statements are equivalent:
(a) $𝓝(ℓ_{p}⊕ C(K₁),ℓ_{q}⊕ C(K₂))$ and $𝓝(ℓ_{p}⊕ C(K₃),ℓ_{q}⊕ C(K₄))$ are isomorphic.
(b) C(K₂) is isomorphic to C(K₄).
Słowa kluczowe
  • Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
  • LATP, Université D'Aix-Marseille, CNRS, 13397 Marseille Cedex 20, France
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