Complemented subspaces of Banach spaces $C(K\times L)$

TL;DR

This paper investigates the structure of complemented subspaces within Banach spaces of continuous functions on product spaces, revealing conditions under which certain classical spaces are complemented and providing counterexamples in the pointwise topology.

Contribution

It establishes new conditions for the existence of complemented subspaces isometric to $C(G)$ in $C(K_1 imes K_2)$ and answers open questions regarding complemented copies of $C([0,1]^"kappa$) and $C(K)$ spaces.

Findings

$C(K_1 imes K_2)$ contains complemented copies of $C(G)$ when $K_1,K_2$ map onto $G$

$C(eta extomega imes eta extomega)$ contains complemented copies of $C([0,1]^ ext{kappa})$ for all $ ext{kappa}",

$C_p(eta extomega imes eta extomega)$ does not contain complemented copies of $C_p(2^ extomega)$.

Abstract

We prove that, for every compact spaces $K_1,K_2$ and compact group $G$, if both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space $C(K_1 \times K_2)$ contains a complemented subspace isometric to the Banach space $C(G)$. Consequently, $C(K_1\times K_2)$ contains a complemented copy of $C([0,1])$ for every non-scattered $K_1,K_2$. Also, answering a question of Alspach and Galego, we get that $C(\beta\omega\times\beta\omega)$ contains a complemented copy of $C([0,1]^\kappa)$ for every cardinal number $1\le\kappa\le{\mathfrak c}$ and hence a complemented copy of $C(K)$ for every metric compact space $K$. On the other hand, for the pointwise topology, we show that $C_p(\beta\omega\times\beta\omega)$ contains no complemented copy of $C_p(2^\omega)$.