The three-space problem for 𝐿¹

Talagrand, Michel

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

The three-space problem for $L^ 1$
HTML articles powered by AMS MathViewer

by Michel Talagrand

J. Amer. Math. Soc. 3 (1990), 9-29

DOI: https://doi.org/10.1090/S0894-0347-1990-1013926-7

PDF | Request permission

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.

References

David J. Aldous, Subspaces of $L^{1}$, via random measures, Trans. Amer. Math. Soc. 267 (1981), no. 2, 445–463. MR 626483, DOI 10.1090/S0002-9947-1981-0626483-2
J. Bourgain, Embedding $L^{1}$ in $L^{1}/H^{1}$, Trans. Amer. Math. Soc. 278 (1983), no. 2, 689–702. MR 701518, DOI 10.1090/S0002-9947-1983-0701518-9
J. Bourgain and H. P. Rosenthal, Martingales valued in certain subspaces of $L^{1}$, Israel J. Math. 37 (1980), no. 1-2, 54–75. MR 599302, DOI 10.1007/BF02762868
J. Bourgain, H. P. Rosenthal, and G. Schechtman, An ordinal $L^{p}$-index for Banach spaces, with application to complemented subspaces of $L^{p}$, Ann. of Math. (2) 114 (1981), no. 2, 193–228. MR 632839, DOI 10.2307/1971293
J. Bourgain and H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), no. 2, 149–188. MR 707202, DOI 10.1016/0022-1236(83)90080-0
P. Enflo and T. W. Starbird, Subspaces of $L^{1}$ containing $L^{1}$, Studia Math. 65 (1979), no. 2, 203–225. MR 557491, DOI 10.4064/sm-65-2-203-225
Hicham Fakhoury, Représentations d’opérateurs à valeurs dans $L^{1}(X,\,\Sigma ,\,\mu )$, Math. Ann. 240 (1979), no. 3, 203–212 (French). MR 526843, DOI 10.1007/BF01362310
Gilles Godefroy, Sous-espaces bien disposés de $L^{1}$-applications, Trans. Amer. Math. Soc. 286 (1984), no. 1, 227–249 (French, with English summary). MR 756037, DOI 10.1090/S0002-9947-1984-0756037-1
Nassif Ghoussoub and Stanisław Szarek, A note on sign-preserving operators on $L^{1}$, Seminar on the geometry of Banach spaces (Paris, 1982) Publ. Math. Univ. Paris VII, vol. 16, Univ. Paris VII, Paris, 1983, pp. 133–138. MR 731951
N. Ghoussoub and H. P. Rosenthal, Martingales, $G_{\delta }$-embeddings and quotients of $L_{1}$, Math. Ann. 264 (1983), no. 3, 321–332. MR 714107, DOI 10.1007/BF01459128
M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
N. J. Kalton, The endomorphisms of $L_{p}(0\leq p\leq i)$, Indiana Univ. Math. J. 27 (1978), no. 3, 353–381. MR 470670, DOI 10.1512/iumj.1978.27.27027
S. V. Kisljakov, Spaces with a “small” annihilator, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 65 (1976), 192–195, 209 (Russian, with English summary). Investigations on linear operators and the theory of functions, VII. MR 0500098
J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Functional Analysis 8 (1971), 225–249. MR 0291772, DOI 10.1016/0022-1236(71)90011-5
Heinrich P. Lotz, N. T. Peck, and Horacio Porta, Semi-embeddings of Banach spaces, Proc. Edinburgh Math. Soc. (2) 22 (1979), no. 3, 233–240. MR 560985, DOI 10.1017/S0013091500016394
J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR 0402915
Aleksander Pełczyński, Banach spaces of analytic functions and absolutely summing operators, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 30, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at Kent State University, Kent, Ohio, July 11–16, 1976. MR 0511811, DOI 10.1090/cbms/030
Gilles Pisier, Une nouvelle classe d’espaces de Banach vérifiant le théorème de Grothendieck, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, x, 69–90 (French, with English summary). MR 487403
James W. Roberts, Representing measures in compact groupoids, Illinois J. Math. 19 (1975), 277–291. MR 377471
Haskell P. Rosenthal, On subspaces of $L^{p}$, Ann. of Math. (2) 97 (1973), 344–373. MR 312222, DOI 10.2307/1970850

Embedding of

in

H. P. Rosenthal, Sign-embeddings of $L^{1}$, Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981) Lecture Notes in Math., vol. 995, Springer, Berlin, 1983, pp. 155–165. MR 717232, DOI 10.1007/BFb0061892