Geodesic
The three-space problem for 𝐿¹
Journal of the American Mathematical Society, Tome 03 (1990) no. 1, pp. 9-29 Cet article a éte moissonné depuis la source American Mathematical Society

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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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Talagrand, Michel. The three-space problem for 𝐿¹. Journal of the American Mathematical Society, Tome 03 (1990) no. 1, pp. 9-29. doi: 10.1090/S0894-0347-1990-1013926-7

[1] Aldous, David J. Subspaces of 𝐿¹, via random measures Trans. Amer. Math. Soc. 1981 445 463

[2] Bourgain, J. Embedding 𝐿¹ in 𝐿¹/𝐻¹ Trans. Amer. Math. Soc. 1983 689 702

[3] Bourgain, J., Rosenthal, H. P. Martingales valued in certain subspaces of 𝐿¹ Israel J. Math. 1980 54 75

[4] Bourgain, J., Rosenthal, H. P., Schechtman, G. An ordinal 𝐿^{𝑝}-index for Banach spaces, with application to complemented subspaces of 𝐿^{𝑝} Ann. of Math. (2) 1981 193 228

[5] Bourgain, J., Rosenthal, H. P. Applications of the theory of semi-embeddings to Banach space theory J. Funct. Anal. 1983 149 188

[6] Enflo, P., Starbird, T. W. Subspaces of 𝐿¹ containing 𝐿¹ Studia Math. 1979 203 225

[7] Fakhoury, Hicham Représentations d’opérateurs à valeurs dans 𝐿¹(𝑋,Σ,𝜇) Math. Ann. 1979 203 212

[8] Godefroy, Gilles Sous-espaces bien disposés de 𝐿¹-applications Trans. Amer. Math. Soc. 1984 227 249

[9] Ghoussoub, Nassif, Szarek, Stanisław A note on sign-preserving operators on 𝐿¹ 1983 133 138

[10] Ghoussoub, N., Rosenthal, H. P. Martingales, 𝐺_{𝛿}-embeddings and quotients of 𝐿₁ Math. Ann. 1983 321 332

[11] Kadec, M. I., Pełczyński, A. Bases, lacunary sequences and complemented subspaces in the spaces 𝐿_{𝑝} Studia Math. 1961/62 161 176

[12] Kalton, N. J. The endomorphisms of 𝐿_{𝑝}(0≤𝑝≤𝑖) Indiana Univ. Math. J. 1978 353 381

[13] Kisljakov, S. V. Spaces with a “small” annihilator Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 1976

[14] Lindenstrauss, J., Pełczyński, A. Contributions to the theory of the classical Banach spaces J. Functional Analysis 1971 225 249

[15] Lotz, Heinrich P., Peck, N. T., Porta, Horacio Semi-embeddings of Banach spaces Proc. Edinburgh Math. Soc. (2) 1979 233 240

[16] Neveu, J. Discrete-parameter martingales 1975

[17] Pełczyński, Aleksander Banach spaces of analytic functions and absolutely summing operators 1977

[18] Pisier, Gilles Une nouvelle classe d’espaces de Banach vérifiant le théorème de Grothendieck Ann. Inst. Fourier (Grenoble) 1978

[19] Roberts, James W. Representing measures in compact groupoids Illinois J. Math. 1975 277 291

[20] Rosenthal, Haskell P. On subspaces of 𝐿^{𝑝} Ann. of Math. (2) 1973 344 373

[21] Rosenthal, H. P. Sign-embeddings of 𝐿¹ 1983 155 165

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