Szlenk and w⁎-dentability indices of C(K)

Abstract

Abstract. Given any compact, Hausdorff space K and 1 < p < ∞, we compute the Szlenk and w * -dentability indices of the spaces C(K) and L p (C(K)). We show that if K is compact, Hausdorff, scattered, CB(K) is the Cantor-Bendixson index of K, and ξ is the minimum ordinal such that CB(K) ω ξ , then Sz(C(K)) = ω ξ and Dz(C(K)) = Sz(L p (C(K))) = ω 1+ξ .

Isomorphisms of $${\mathcal {C}}(K, E)$$ Spaces and Height of K

“…For example, Γ(ht [22]. On the other hand, the condition Γ(ht(K 1 )) = Γ(ht(K 2 )) still remains necessary for the spaces C(K 1 ) and C(K 2 ) to be isomorphic, which follows from the computation of the Szlenk index of a C(K) space due to Causey [9], and the fact that the Szlenk index is preserved by isomorphisms of Banach spaces. Now we turn our attention to case of spaces of vector-valued functions.…”

Subprojectivity of Projective Tensor Products of Banach Spaces of Continuous Functions

“…Our proof makes explicit the Grasberg norm, which has previously only appeared implicitly. Previously a number of results known for C(K) spaces with K countable, compact, Hausdorff (such as those in [16], [10], and [7]) have been extended to the general case of scattered, compact, Hausdorff spaces (see [1], [2]), and the decomposition implicit in the Grasberg norm has been additionally used in, for example, [3] and [5]. Many of the aforementioned proofs in the case of K countably infinite used the fact that C(K) is isomorphic to C(ω ω ξ ) for some countable ξ , which provides a convenient basis for induction.…”