The number of closed ideals in the algebra of bounded operators on Lebesgue spaces

Abstract

We survey some of the recent progress in determining the number of two-sided closed ideals in the Banach algebras of bounded linear operators on Lebesgue spaces, $L_p[0,1]$. In particular, we discuss two recent results: the first of Johnson, Pisier, and the author, showing that there are a continuum of such ideal in the case of $p=1$; the second a result of Johnson and the author, showing that in the case $1<p<\infty$, $p\ne 2$, there are exactly 2 to the continuum such ideals.