Preservation of primariness under $\ell_1$-, $c_0$-, and $\ell_\infty$-sums of Banach spaces

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 $X^*$, 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 $1\leq p<\infty$ 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 $C[0,1]^*$ has the UPFP and is primary.