Copies of $c_{0}(Γ)$ in $C(K, X)$ spaces

Abstract

Abstract. We extend some results of Rosenthal, Cembranos, Freniche, E. Saab-P. Saab and Ryan to study the geometry of copies and complemented copies of c 0 (Γ) in the classical Banach spaces C(K, X) in terms of the cardinality of the set Γ, of the density and caliber of K and of the geometry of X and its dual space X * . Here are two sample consequences of our results:(1) If C([0, 1], X) contains a copy of c 0 (ℵ 1 ), then X contains a copy of c 0 (ℵ 1 ).(2) C(βN, X) contains a complemented copy of c 0 (ℵ 1 ) if and only if X contains a copy of c 0 (ℵ 1 ). Some of our results depend on set-theoretic assumptions. For example, we prove that it is relatively consistent with ZFC that if C(K) contains a copy of c 0 (ℵ 1 ) and X has dimension ℵ 1 , then C(K, X) contains a complemented copy of c 0 (ℵ 1 ).

On complemented copies of $c_0(\omega _1)$ in $C(K^n)$ spaces

“…This notion plays a crucial role in the result of Galego and Hagler and of Todorcevic mentioned above ( [8,20]). Theorem 1.1 ( [5,8]). Let X be a Banach space.…”

“…If X is a Banach space, then a sequence (x * ξ ) ξ<ω1 of elements of the unit sphere of X * is called an ω 1 -Josefson-Nissenzweig sequence if and only if (x * ξ (x)) ξ<ω1 belongs to c 0 (ω 1 ) for any x ∈ X. This notion plays a crucial role in the result of Galego and Hagler and of Todorcevic mentioned above ( [8,20]). Theorem 1.1 ( [5,8]).…”

When is complemented in tensor products of ?

“…More precisely, we will prove that for every Banach space X , infinite set I , and infinite cardinal τ, one has Additionally, if cf, then one also has Remark The equivalence cannot be extended to the case . Setting and , by [, Theorem 5.3] we know that although by [, Corollary 11, p. 156].…”

“…Remark is optimal for every infinite regular cardinal κ. Indeed, setting , again by [, Theorem 4.5] we have but once again by [, Corollary 11, p. 156].…”

Complementations in $C(K,X)$ and $\ell_\infty(X)$

“…The first part of our investigation concerns C(K, X) spaces and goes back to the classical and celebrated Cembranos-Freniche theorem [7] which states that C(K, X) has a complemented subspace isomorphic to c 0 whenever K is an infinite Hausdorff compactum and X is an infinite dimensional Banach space. The Cembranos-Freniche theorem in the past decades has influenced many lines of research, for recent examples [4] [9] [10], and was extended in many directions [27] [26] [14]. Galego and Hagler in [14], among other results, isolated conditions on K and X implying that C(K, X) will have a complemented subspace isomorphic to c 0 (Γ ), where Γ is an infinite set, not necessarily countable.…”