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Abstract The classical Banach space $$L_1(L_p)$$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $$L_1(L_p) (1 < p < \infty )$$ is primary, meaning that whenever $$L_1(L_p) = E\oplus F$$ , where E and F are closed subspaces of $$L_1(L_p)$$ , 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. 

Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by $${(e_{(k,j)}^*)_j}$$ , for $$k\in\N$$ , let $$Z=\ell^\infty(X_k:k\kin\N)$$ be their l ∞ -sum, and let $$T:Z\to Z$$ be a bounded linear operator with a large diagonal, i.e. , $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive. 

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most 6. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into l1. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterisations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterisation, in terms of a sequence of graphs with uniformly bounded degree, of the notion of trivial Bourgain–Milman–Wolfson type for arbitrary metric spaces, similar to Ostrovskii’s characterisation previously obtained in Ostrovskii (C. R. Acad. Bulgare Sci. 64(6), 775–784 (2011)).