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

Elói Medina Galego, and Christian Samuel. "The classical subspaces of the projective tensor products of $ℓ_{p}$ and C(α) spaces, α < ω₁." Studia Mathematica 214.3 (2013): 237-250. <http://eudml.org/doc/286180>.

@article{ElóiMedinaGalego2013,
abstract = {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).},
author = {Elói Medina Galego, Christian Samuel},
journal = {Studia Mathematica},
keywords = {projective tensor products; isomorphic classifications; spaces of nuclear operators},
language = {eng},
number = {3},
pages = {237-250},
title = {The classical subspaces of the projective tensor products of $ℓ_\{p\}$ and C(α) spaces, α < ω₁},
url = {http://eudml.org/doc/286180},
volume = {214},
year = {2013},
}

TY - JOUR
AU - Elói Medina Galego
AU - Christian Samuel
TI - The classical subspaces of the projective tensor products of $ℓ_{p}$ and C(α) spaces, α < ω₁
JO - Studia Mathematica
PY - 2013
VL - 214
IS - 3
SP - 237
EP - 250
AB - 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).
LA - eng
KW - projective tensor products; isomorphic classifications; spaces of nuclear operators
UR - http://eudml.org/doc/286180
ER -