This is an announcement for the paper ``The number of closed ideals in $L(L_p)$” by William B. Johnsonhttps://arxiv.org/search/math?searchtype=author&query=Johnson%2C+W+B, Gideon Schechtmanhttps://arxiv.org/search/math?searchtype=author&query=Schechtman%2C+G.

Abstract: We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\not= 2<\infty$. 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 $2^{2^{\aleph_0}}$ closed ideals in term 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 ${\frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.

https://arxiv.org/abs/2003.11414