The space $L_1(L_p)$ is primary for $1<p<\infty$

Abstract

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$\Vert f\Vert ={\int }_0^1({\int }_0^1|f(x,y){|}^{p}dy{)}^{1/p}dx<\infty .$$ We show that $L_1(L_p)$ $(1<p<\infty )$ is primary, meaning that, whenever $L_1(L_p)=E\oplus F$ then either $E$ or $F$ is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary, for a large class of rearrangement invariant Banach function spaces.

43 pages