Extension operators on balls and on spaces of finite sets

TL;DR

This paper investigates extension operators between certain subset spaces and demonstrates their non-existence between scaled unit balls of nonseparable Hilbert spaces under the weak topology.

Contribution

It introduces new results on the non-existence of extension operators between scaled unit balls in nonseparable Hilbert spaces.

Findings

No extension operator exists between $C(\lambda B_H)$ and $C(\mu B_H)$ for $0<\lambda<\mu$ in nonseparable Hilbert spaces.

Provides insights into the structure of extension operators on spaces of finite sets and their limitations.

Highlights the differences in extension operator behavior in infinite-dimensional Hilbert spaces.

Abstract

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 $T: C(\lambda B_H)\to C(\mu B_H)$.