Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, and Elói Medina Galego. "Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K." Studia Mathematica 207.2 (2011): 153-180. <http://eudml.org/doc/285479>.

@article{DaleE2011,
abstract = {A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C(ω^\{ω\})$ then X contains a copy of c₀. Moreover, we show that $C(ω^\{ω\})$ is not even a quotient of $C(βℕ ×[1,ω],ℓ_\{p\})$, 1 < p < ∞. We then completely determine the separable C(K) spaces which are isomorphic to a complemented subspace or a quotient of a $C(βℕ ×[1,α],ℓ_\{p\})$ space for countable ordinals α and 1 ≤ p < ∞. As a consequence, we obtain the isomorphic classification of the $C(βℕ ×K,ℓ_\{p\})$ spaces for infinite compact metric spaces K and 1 ≤ p < ∞. Indeed, we establish the following more general cancellation law. Suppose that the Banach space X contains no copy of c₀ and K₁ and K₂ are infinite compact metric spaces, then the following statements are equivalent: (1) C(βℕ ×K₁,X) is isomorphic to C(βℕ ×K₂,X). (2) C(K₁) is isomorphic to C(K₂). These results are applied to the isomorphic classification of some spaces of compact operators.},
author = {Dale E. Alspach, Elói Medina Galego},
journal = {Studia Mathematica},
keywords = {isomorphisms of spaces; Stone-Čech compactification; compact metric spaces},
language = {eng},
number = {2},
pages = {153-180},
title = {Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K},
url = {http://eudml.org/doc/285479},
volume = {207},
year = {2011},
}

TY - JOUR
AU - Dale E. Alspach
AU - Elói Medina Galego
TI - Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K
JO - Studia Mathematica
PY - 2011
VL - 207
IS - 2
SP - 153
EP - 180
AB - A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C(ω^{ω})$ then X contains a copy of c₀. Moreover, we show that $C(ω^{ω})$ is not even a quotient of $C(βℕ ×[1,ω],ℓ_{p})$, 1 < p < ∞. We then completely determine the separable C(K) spaces which are isomorphic to a complemented subspace or a quotient of a $C(βℕ ×[1,α],ℓ_{p})$ space for countable ordinals α and 1 ≤ p < ∞. As a consequence, we obtain the isomorphic classification of the $C(βℕ ×K,ℓ_{p})$ spaces for infinite compact metric spaces K and 1 ≤ p < ∞. Indeed, we establish the following more general cancellation law. Suppose that the Banach space X contains no copy of c₀ and K₁ and K₂ are infinite compact metric spaces, then the following statements are equivalent: (1) C(βℕ ×K₁,X) is isomorphic to C(βℕ ×K₂,X). (2) C(K₁) is isomorphic to C(K₂). These results are applied to the isomorphic classification of some spaces of compact operators.
LA - eng
KW - isomorphisms of spaces; Stone-Čech compactification; compact metric spaces
UR - http://eudml.org/doc/285479
ER -