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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