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The number of closed ideals in L(L_p)

William B. Johnson Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A. Gideon Schechtman Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

Functional Analysis Spectral Theory and Operator Algebra mathscidoc:2203.12001

Acta Mathematica, 227, (1), 103-113, 2021.11
We show that there are 2^{2^{ℵ_0}} different closed ideals in the Banach algebra L(L_p(0,1)),1<p≠2<∞. This solves a problem in A. Pietsch’s 1978 book “Operator Ideals”. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non-Hilbertian. We give a criterion for a space with an unconditional basis to have 2^{2^{ℵ_0}} closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for 1<q<2 the space Xq of Rosenthal, which is isomorphic to a complemented subspace of L_q(0,1), admits such an operator.
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@inproceedings{william2021the,
  title={The number of closed ideals in L(L_p)},
  author={William B. Johnson, and Gideon Schechtman},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310113159809140924},
  booktitle={Acta Mathematica},
  volume={227},
  number={1},
  pages={103-113},
  year={2021},
}
William B. Johnson, and Gideon Schechtman. The number of closed ideals in L(L_p). 2021. Vol. 227. In Acta Mathematica. pp.103-113. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310113159809140924.
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