Ideals in $L(L_1)$

William B.Johnson; Gilles Pisier; Gideon Schechtman

arXiv:1811.06571·math.FA·November 14, 2019·1 cites

Ideals in $L(L_1)$

William B.Johnson, Gilles Pisier, Gideon Schechtman

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TL;DR

This paper demonstrates the existence of a continuum of closed ideals in the Banach algebra of bounded linear operators on various classical spaces, answering a longstanding open question.

Contribution

It proves that $L(L_1)$, $L(C[0,1])$, and $L(ell_ re each equipped with a continuum of closed ideals, resolving a question from 1978.

Findings

01

$L(L_1)$ has a continuum of closed ideals.

02

$L(C[0,1])$ contains a continuum of closed ideals.

03

$L(ell_)$ also has a continuum of closed ideals.

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 $L (ℓ_{\infty})$ has a continuum of closed ideals.

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Taxonomy

TopicsRings, Modules, and Algebras