On Pełczyński's problem concerning the uniqueness of symmetric structure

Let $E$ and $F$ be separable symmetric Banach sequence spaces, and let $C_E$ and $C_F$ be the corresponding ideals of compact operators on a separable Hilbert space. We prove that if $C_F$ is isomorphic to a complemented subspace of $S_E$, then $F={\mathrm{\ell }}_2$. As a consequence, if $C_E$ is isomorphic to $C_F$, then $E=F$ with equivalent norms. This answers Pełczyński’s 1979 question concerning the uniqueness of symmetric structure. We also strengthen Arazy’s result by showing that if $E$ has finite cotype and $C_F$ embeds isomorphically into $S_E$, then $F={\mathrm{\ell }}_2$. For reflexive $F\mathrm{\ne }{\mathrm{\ell }}_2$ with non-trivial Boyd indices, the embedding theorems of Lindenstrauss and Szankowski yield a separable reflexive symmetric sequence space $E$ with trivial cotype such that $C_F$ embeds isomorphically into $S_E$. This gives negative answers to several questions raised by Arazy in 1981.