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2011 | 207 | 2 | 153-180
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Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

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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.
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  • Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A.
  • Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-2-4
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