The three-space problem for 𝐿¹

Talagrand, Michel

doi:10.1090/S0894-0347-1990-1013926-7

The three-space problem for $L^ 1$

Author: Michel Talagrand
Journal: J. Amer. Math. Soc. 3 (1990), 9-29
MSC: Primary 46E30; Secondary 46B20
DOI: https://doi.org/10.1090/S0894-0347-1990-1013926-7
MathSciNet review: 1013926
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Abstract: We construct a subspace $X$ of ${L^1}$ such that $X$ is an ${l^1}$-sum of spaces isomorphic to ${l^1}$ but such that ${L^1}/X$ does not contain a copy of ${L^1}$. We also construct two Banach spaces ${E_1}$, ${E_2}$ that do not contain a copy of ${L^1}$ but such that ${E_1} \times {E_2}$ contains a copy of ${L^1}$. Moreover, the projections of ${L^1}$ on each factor are one-to-one, and the images of the unit ball of ${L^1}$ are closed. These examples settle questions of J. Lindenstrauss, P. Pelczynski, J. Bourgain, and H. P. Rosenthal.

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