Topological group actions by group automorphisms and Banach representations

TL;DR

This paper investigates when topological group actions by automorphisms can be represented on various Banach spaces, revealing limitations and possibilities for low-complexity representations, including counterexamples.

Contribution

It characterizes Banach representability of conjugation actions and provides counterexamples for certain groups and actions, highlighting differences from standard group representations.

Findings

Conjugation actions are always Banach representable on some Banach space.

Certain conjugation actions are not reflexively or Asplund representable, even for classical groups.

Some actions are representable on Rosenthal Banach spaces, but not on Asplund spaces.

Abstract

We study Banach representability for actions of topological groups on groups by automorphisms (in particular, actions of groups on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group $G$ on itself, the conjugation action need not be reflexively representable even for $\mathrm{SL}_2(\mathbb{R})$. The conjugation action of $\mathrm{SL}_n(\mathbb{R})$ is not Asplund representable for every $n \geq 4$. The linear action of $\mathrm{GL}(n,\mathbb{R})$ on $\mathbb{R}^n$, for every $n \geq 2$, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of $l_1$). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for $\mathrm{SL}_2(\mathbb{R})$. As a byproduct we obtain some counterexamples about Banach representations of homogeneous $G$-actions $G/H$.