Continuous Linear Surjections from $C_p(X)$ onto Symmetric Sequence Ideals in $c_0$

Rosenthal’s classical theorem says that, for every infinite compact space $X$, the Banach space $C(X)$ admits a quotient isomorphic to either $c_0$ or ${\mathrm{\ell }}_2$. The corresponding question for $C_p(X)$, the space $C(X)$ endowed with the topology of pointwise convergence, is much subtler and still open; the only compact spaces for which the existence of an infinite-dimensional metrizable quotient is not presently settled in ZFC are Efimov compacta. We prove that the Banach space case cannot be reproduced with the usual sequence spaces carrying their pointwise topologies: Let $X$ be a Tychonoff space and let $E\subseteq c_0$ be a non-trivial symmetric sequence ideal endowed with the subspace topology inherited from ${\mathbb{R}}^{\mathbb{N}}$. Then the existence of a continuous linear surjection $T:C_p(X)\to E_p$ implies $E=c_0$, where $E_p$ means $E$ with the topology inherited from ${\mathbb{R}}^{\mathbb{N}}$. Hence, no proper non-zero symmetric sequence ideal of $c_0$ can be realized as a continuous linear image of a $C_p$-space. Combining this result with the characterization of the Josefson–Nissenzweig property for $C_p(X)$ obtained by Banakh, K\k{a}kol, and '{S}liwa, we derive a complete characterization of all pairs $(X,E)$ for which such a surjection exists. In particular, for every $0<q<\mathrm{\infty }$, there is no continuous linear surjection $C_p(X)\to ({\mathrm{\ell }}_q{)}_p$.