The classical subspaces of the projective tensor products of $\ell _{p}$ and $C(\alpha )$ spaces, $\alpha < \omega _{1}$

Tom 214 / 2013

Elói Medina Galego, Christian Samuel Studia Mathematica 214 (2013), 237-250 MSC: Primary 46B03; Secondary 46B25. DOI: 10.4064/sm214-3-3


We completely determine the $\ell_{q}$ and $C(K)$ spaces which are isomorphic to a subspace of $\def\widot{\mathbin{\widehat{\otimes}}}\ell_{p} \widot_{\pi} C(\alpha)$, the projective tensor product of the classical $\ell_{p}$ space, $1 \leq p< \infty$, and the space $C(\alpha)$ of all scalar valued continuous functions defined on the interval of ordinal numbers $[1, \alpha]$, $\alpha< \omega_{1}$. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from $\ell_{p}$ to $\ell_{1}$, $1 \leq p< \infty$.

The first main theorem is an extension of a result of E. Oja and states that the only $\ell_{q}$ space which is isomorphic to a subspace of $\def\widot{\mathbin{\widehat{\otimes}}}\ell_{p} \widot_{\pi} C(\alpha)$ with $1 \leq p\leq q < \infty$ and $\omega \leq \alpha< \omega_{1}$ is $\ell_{p}$. 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 $\mathcal{N}(X, Y)$ of nuclear operators, where $X$ and $Y$ are direct sums of $\ell_{p}$ and $C(K)$ spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that $K_{1}$ and $K_{3}$ are finite or countable compact metric spaces of the same cardinality and $1 (a) $\mathcal{N}(\ell_{p} \oplus C(K_{1}), \ell_{q} \oplus C(K_{2}))$ and $\mathcal{N}(\ell_{p} \oplus C(K_{3}), \ell_{q} \oplus C(K_{4}))$ are isomorphic.

(b) $C(K_{2})$ is isomorphic to $C(K_{4})$.


  • Elói Medina GalegoDepartment of Mathematics
    University of São Paulo
    São Paulo, Brazil 05508-090
  • Christian SamuelLATP
    Université D'Aix-Marseille, CNRS
    13397 Marseille Cedex 20, France

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