Let $1\leq p<\infty.$ Denote $S_p(\ell_2)$ be the set of all compact operator $x$ on $\ell_2$ such that $Tr(|x|^p)<\infty.$ Define $\|x\|_{S_p(\ell_2)}:=Tr(|x|^p)^{\frac{1}{p}}.$ This makes $S_p(\ell_2)$ a Banach space. What is the largest closed two-sided ideal in the Banach algebra of set of all bounded linear maps on $S_p(\ell_2)$?
