Ideals in \(L(L_1)\)

  • W. B. JohnsonEmail author
  • G. Pisier
  • G. Schechtman


The main result is that there are infinitely many; in fact, a continuum; of closed (two-sided) ideals in the Banach algebra \(L(L_1)\) of bounded linear operators on \(L_1(0,1)\). This answers a question from Pietsch’s 1978 book “Operator Ideals”. The proof also shows that L(C[0, 1]) contains a continuum of closed ideals. Finally, a duality argument yields that \(L(\ell _\infty )\) has a continuum of closed ideals.

Mathematics Subject Classification

Primary 46B03 46B20 Secondary 46B28 



  1. 1.
    Albiac, F., Kalton, NJ: Topics in Banach space theory. In: Graduate Texts in Mathematics, vol. 233. Springer, New York (2006)Google Scholar
  2. 2.
    Astashkin, S.V., Hernández, F., Semenov, E.M.: Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces. Stud. Math. 193(3), 269–283 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bennett, G., Dor, L.E., Goodman, V., Johnson, W.B., Newman, C.M.: On uncomplemented subspaces of \(L_{p},\) \(1<p<2\). Israel J. Math. 26(2), 178–187 (1977)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bennett, G., Goodman, V., Newman, C.M.: Norms of random matrices. Pac. J. Math. 59(2), 359–365 (1975)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Berkson, E., Porta, H.: Representations of \({\cal{B}}(X)\). J. Funct. Anal. 3, 1–34 (1969)CrossRefGoogle Scholar
  6. 6.
    Bessaga, C., Pełczyński, A.: Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions. Stud. Math. 19, 53–62 (1960)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bourgain, J.: Bounded orthogonal systems and the \(\Lambda (p)\)-set problem. Acta Math. 162(3–4), 227–245 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bourgain, J., Rosenthal, H.P., Schechtman, G.: An ordinal \(L_p\)-index for Banach spaces, with application to complemented subspaces of \(L_p\). Ann. Math. (2) 114(2), 193–228 (1981)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Calkin, J.: Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. Math. 42(4), 39–873 (1941)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Diestel, J., Uhl, J.J. Jr.: Vector Measures. With a foreword by B. J. Pettis. Mathematical Surveys, No. 15. American Mathematical Society, Providence (1977)Google Scholar
  11. 11.
    Enflo, P., Starbird, T.W.: Subspaces of \(L^1\) containing \(L^1\). Stud. Math. 65(2), 203–225 (1979)CrossRefGoogle Scholar
  12. 12.
    Gohberg, I.C., Markus, A.S., Feldman, I.A.: Normally solvable operators and ideals associated with them. (English translation). Am. Math. Soc. Transl. 61, 63–84 (1967). Russian original appearing in Bul. Akad. Štiince RSS Moldoven 10 (76), 51–70 (1960)Google Scholar
  13. 13.
    Johnson, W.B., Maurey, B., Schechtman, G., Tzafriri, L.: Symmetric structures in Banach spaces. Mem. Am. Math. Soc. 19(217), v+298 (1979)MathSciNetzbMATHGoogle Scholar
  14. 14.
  15. 15.
    Lewis, D.R., Stegall, C.: Banach spaces whose duals are isomorphic to \(\ell _1(\Gamma )\). J. Funct. Anal. 12, 177–187 (1973)CrossRefGoogle Scholar
  16. 16.
    Mankiewicz, P.: A superreflexive Banach space X with \(L(X)\) admitting a homomorphism onto the Banach algebra \(C(\beta {\mathbb{N}})\). Israel J. Math. 65(1), 1–16 (1989)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Miljutin, A. A. Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum. (Russian) Teor. Funkcional. Anal. i Prilozen. Vyp. 2, 150–156 (1966)Google Scholar
  18. 18.
    Pietsch, A.: Operator Ideals. Volume 16 of Mathematische Monographien. Deutscher Verlag der Wissenschaften (1978)Google Scholar
  19. 19.
    Rosenthal, H.P.: On the subspaces of \(L^{p}\) \((p>2)\) spanned by sequences of independent random variables. Israel J. Math. 8, 273–303 (1970)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rosenthal, H.P.: On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from \(L^p(\mu )\) to \(L^r(\nu )\). J. Funct. Anal. 4, 176–214 (1969)CrossRefGoogle Scholar
  21. 21.
    Schechtman, G.: Examples of \(\cal{L}_p\)-spaces (\(1 < p\ne 2 < \infty \)). Israel J. Math. 22, 138–147 (1975)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schlumprecht, T., Zsák, A.: The algebra of bounded linear operators on \(\ell _p\oplus \ell _q\) has infinitely many closed ideals. J. Reine Angew. Math. 735, 225–247 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Whitley, R.J.: Strictly singular operators and their conjugates. Trans. Am. Math. Soc. 113, 252–261 (1964)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations