The classical subspaces of the projective tensor products of $\ell _{p}$ and $C(\alpha )$ spaces, $\alpha < \omega _{1}$
Volume 214 / 2013
Abstract
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
(b) $C(K_{2})$ is isomorphic to $C(K_{4})$.
