Primariness of the spaces ${\ell }_p(C(K))$ for $1\le p\le \infty$

Abstract

We prove that the spaces ${\ell }_p(C(\alpha ))$ and ${\ell }_p(C[0,1])$ have the uniform primary factorisation property whenever $\alpha$ is an ordinal and $1<p\le \infty$. 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 $1\le p\le \infty$.