The number of closed ideals in $L(L_p)$

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 A. Pietsch's 1978 book Operator Ideals. The proof is quite from other methods of producing space bounded operators on space; particular, not contained strictly singular and yet do contain projections onto subspaces non Hilbertian. give criterion for with an unconditional basis to have terms existence single operator some special asymptotic properties. then $1<q<2$ ${\frak X}_q$ Rosenthal, which isomorphic complemented subspace $L_q(0,1)$, admits such operator.