Ideals in $L(L_1)$

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.

Final version, will appear in Mathematische Annalen