Preservation of primariness under ${\ell }_1$-, $c_0$-, and ${\ell }_{\infty }$-sums of Banach spaces

Abstract

We prove transfer principles for the uniform primary factorisation property (UPFP) from a Banach space $X$ to the vector-valued sequence spaces ${\ell }_1(X)$, $c_0(X)$ and ${\ell }_{\infty }(X)$. The hypotheses are either finite-cotype assumptions on $X$ or $X^{\ast }$, or natural self-similarity assumptions on $X$. Consequently, under these conditions, the resulting vector-valued sequence spaces are primary. As applications, we recover the primariness of ${\ell }_{\infty }(L_p)$ for $1\le p<\infty$ without using Bourgain's localisation method, and obtain the primariness of $c_0(L_1)$. We also show that ${\ell }_1(\Gamma ,L_1[0,1])$ has the UPFP for every set $\Gamma$, and consequently that $C[0,1{]}^{\ast }$ has the UPFP and is primary.