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

Elói Medina Galego¹ ; Christian Samuel²

¹ Department of Mathematics University of São Paulo São Paulo, Brazil 05508-090
² LATP Université D'Aix-Marseille, CNRS 13397 Marseille Cedex 20, France

Studia Mathematica, Tome 214 (2013) no. 3, pp. 237-250

Résumé

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 $C(K_{4})$.

Détail

DOI : 10.4064/sm214-3-3

Keywords: completely determine ell spaces which isomorphic subspace def widot mathbin widehat otimes ell widot alpha projective tensor product classical ell space leq infty space alpha scalar valued continuous functions defined interval ordinal numbers alpha alpha omega order extend result nbsp tong concerning diagonal block matrices representing operators ell ell leq infty first main theorem extension result oja states only ell space which isomorphic subspace def widot mathbin widehat otimes ell widot alpha leq leq infty omega leq alpha omega ell second main theorem concerning spaces improves result bessaga czy ski which allows classify isomorphism separable spaces mathcal nuclear operators where direct sums ell spaces precisely prove following cancellation law separable banach spaces suppose finite countable compact metric spaces cardinality infty infinite compact metric spaces following statements equivalent mathcal ell oplus ell oplus mathcal ell oplus ell oplus isomorphic isomorphic