Primariness of the spaces $\ell_p(C(K))$ for $1 \leq p \leq \infty$

Abstract

We prove that the spaces $\ell_p(C(α))$ and $\ell_p(C[0,1])$ have the uniform primary factorisation property whenever $α$ is an ordinal and $1<p\leq\infty$. For the case $p=1$, we establish a general criterion ensuring that $\ell_1(X)$ inherits the uniform primary factorisation property from $X$. As a consequence, $\ell_p(C(K))$ is primary for every compact metrizable space $K$ and every $1 \leq p \leq \infty$.