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

TL;DR

This paper explores the structure of certain Banach spaces, specifically $C(K,X)$ and $ ext{ extlbrack} ext{ell}_{ ext{ extlbrack} ext{infty}}(X)$, focusing on complemented subspaces and their isomorphic properties, extending previous results and establishing new classification criteria.

Contribution

It extends known results on the geometry of $C(K,X)$ spaces, characterizes complemented subspaces in $ ext{ extlbrack} ext{ell}_{ ext{ extlbrack} ext{infty}}(X)$, and provides new isomorphism criteria for spaces of continuous functions.

Findings

If $ ext{ extlbrack} ext{ell}_{ ext{ extlbrack} ext{infty}}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then some finite power of $X$ contains a subspace isomorphic to $c_0(Y)$.

Spaces $ ext{ extlbrack} ext{ell}_{ ext{ extlbrack} ext{infty}}(C(K))$ are classified by the cardinality of $K$ when they are isomorphic to their $c_0$-sum.

For infinite metric compacta $K_1, K_2$, the spaces $ ext{ extlbrack} ext{ell}_{ ext{ extlbrack} ext{infty}}(C(K_1))$ and $ ext{ extlbrack} ext{ell}_{ ext{ extlbrack} ext{infty}}(C(K_2))$ are isomorphic if and only if $C(K_1)$ and $C(K_2)$ are isomorphic.

Abstract

We investigate the geometry of $C(K,X)$ and $\ell_{\infty}(X)$ spaces through complemented subspaces of the form $\left(\bigoplus_{i\in \varGamma}X_i\right)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from \cite{AlspachGalego}. On $\ell_{\infty}$-sums of Banach spaces we prove that if $\ell_{\infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n \in \mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: (1) If $C(K)\sim c_0(C(K))$ and $C(L)\sim c_0(C(L))$ and $\ell_{\infty}(C(K))\sim \ell_{\infty}(C(L))$, then $K$ and $L$ have the same cardinality. (2) If $K_1$ and $K_2$ are infinite metric compacta, then $\ell_{\infty}(C(K_1))\sim \ell_{\infty}(C(K_2))$ if and only if $C(K_1)$ is isomorphic to $C(K_2)$.