# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "Topological group actions by group automorphisms and Banach representations"

#### Megrelishvili, Michael

This project is dedicated to Vladimir Pestov on the occasion of his 65th birthday. To every Banach space $V$ one may associate a continuous dual action of the topological group $Iso(V)$ of all linear isometries on the weak-star compact unit ball $B^*$ of the dual space $V^*$. Which actions $G \times X \to X$ are "subactions" of $Iso(V) \times B^* \to B^*$ for nice Banach spaces $V$ ? We study Banach representability for actions of topological groups on groups by automorphisms; in particular, actions on itself by conjugations. 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 $SL_2(R)$. The conjugation action of $SL_n(R)$ is not Asplund representable for every $n > 3$. The linear action of $GL_n(R)$ on $R^n$, for every $n > 1$, 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 $SL_2(R)$. As a byproduct we obtain some counterexamples about Banach representations of homogeneous $G$-actions $G/H$. For more details we refer to arXiv:2110.01386, 2021.

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