Preservation of primariness under ${\mathrm{\ell }}_1$-, $c_0$-, and ${\mathrm{\ell }}_{\mathrm{\infty }}$-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 ${\mathrm{\ell }}_1(X)$, $c_0(X)$ and ${\mathrm{\ell }}_{\mathrm{\infty }}(X)$. 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 ${\mathrm{\ell }}_{\mathrm{\infty }}(L_p)$ for $1\le p<\mathrm{\infty }$ without using Bourgain’s localisation method, and obtain the primariness of $c_0(L_1)$. We also show that ${\mathrm{\ell }}_1(\mathrm{\Gamma },L_1[0,1])$ has the UPFP for every set $\mathrm{\Gamma }$, and consequently that $C[0,1{]}^{\ast }$ has the UPFP and is primary.