The classical subspaces of the projective tensor products of $\ell_p$ and $C(\alpha)$ spaces, $\alpha<\omega_1$
Résumé
We completely determine the ℓq and C(K) spaces which are isomorphic to a subspace of ℓp⊗ˆπC(α), the projective tensor product of the classical ℓp 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 ℓp to ℓ1, 1≤p<∞.
The first main theorem is an extension of a result of E. Oja and states that the only ℓq space which is isomorphic to a subspace of ℓp⊗ˆπC(α) with 1≤p≤q<∞ and ω≤α<ω1 is ℓp. 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 (X,Y) of nuclear operators, where X and Y are direct sums of ℓp 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 (a) (ℓp⊕C(K1),ℓq⊕C(K2)) and (ℓp⊕C(K3),ℓq⊕C(K4)) are isomorphic.