The space $L_1(L_p)$ is primary for $1

Abstract

The classical Banach space L1(Lp) consists of measurable scalar functions f on the unit square for whichWe show that L1(Lp) (1 < p < ∞) is primary, meaning that, whenever L1(Lp) = E ⊕ F then either E or F is isomorphic to L1(Lp). More generally we show that L1(X) is primary, for a large class of rearrangement invariant Banach function spaces. Contents1. Introduction 1.1. Background and History 1.2. The present paper. 2. Preliminaries 2.1. Factors and Projectional Factors up to Approximation 2.2. The Haar system in L 1 2.3. Haar multipliers on L 1 2.4. Haar system spaces 2.5. Complemented subspaces of L 1 (X) isomorphic to L 1 (X) 2.6. Decompositions of operators on L 1 (X) 2.7. Operators on L 1 associated to an operator on L 1 (X) 3. Compactness properties of families of operators 3.1. WOT-sequentially compact families 3.2. Compactness in operator norm 4.