Ideals in $L(L_1)$

William B. Johnson; Gilles Pisier; Gideon Schechtman

Ideals in $L(L_1)$

Functional Analysis 2019-11-14 v2

Authors: William B. Johnson , Gilles Pisier , Gideon Schechtman

Abstract

The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$ . This answers a question from A. 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.

Keywords

operator theory on banach spaces banach space banach spaces

Cite

@article{arxiv.1811.06571,
  title  = {Ideals in $L(L_1)$},
  author = {William B. Johnson and Gilles Pisier and Gideon Schechtman},
  journal= {arXiv preprint arXiv:1811.06571},
  year   = {2019}
}

Comments

Final version, will appear in Mathematische Annalen

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