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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.
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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.more » « less
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Abstract We formulate general conditions which imply that $${\mathcal L}(X,Y)$$ , the space of operators from a Banach space X to a Banach space Y , has $$2^{{\mathfrak {c}}}$$ closed ideals, where $${\mathfrak {c}}$$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in $${\mathcal L}\left (\ell _p\oplus \ell _q\right )$$ is exactly $$2^{{\mathfrak {c}}}$$ for all $$1<\infty $$ .more » « less
