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

TL;DR

This paper proves that the Banach space $L_1(L_p)$ is primary for all $1<p< finite$, meaning it cannot be decomposed into two nontrivial complemented subspaces both not isomorphic to itself.

Contribution

It establishes the primarity of $L_1(L_p)$ spaces and extends this result to a broad class of rearrangement invariant Banach function spaces.

Findings

$L_1(L_p)$ is primary for $1<p< finite$

Primarity extends to many rearrangement invariant Banach spaces

Decomposition into nontrivial complemented subspaces always includes a copy of $L_1(L_p)$

Abstract

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.$$ 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.