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

Elói Medina Galego ; Christian Samuel

Studia Mathematica, Tome 215 (2013), p. 237-250 / Harvested from The Polish Digital Mathematics Library

Résumé

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 (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) $(ℓ_{p}⊕ C(K₁),ℓ_{q}⊕ C(K₂))$ and $(ℓ_{p}⊕ C(K₃),ℓ_{q}⊕ C(K₄))$ are isomorphic. (b) C(K₂) is isomorphic to C(K₄).

Détail

Publié le : 2013-01-01

Zbl 1280.46006

EUDML-ID : urn:eudml:doc:286180