Extension operators on balls and on spaces of finite sets

We study extension operators between spaces ${\sigma }_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0<\lambda <\mu$, there is no extension operator $T: C(\lambda B_H)\to C(\mu B_H)$.