**"Topological group actions by group automorphisms and Banach representations"**
#### Megrelishvili, MichaelThis 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. |