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Journal of Operator Theory

Volume 86, Issue 1, Summer 2021  pp. 51-60.

On the isometrisability of group representations on $p$-spaces

Authors:  Maria Gerasimova (1), Andreas Thom (2)
Author institution: (1) Department of Mathematics, Bar-Ilan University, Ramat Gan, 52900, Israel
(2) Institut fuer Geometrie, TU Dresden, Dresden, 01069, Germany

Summary: In this note we consider a $p$-isometrisability property of discrete groups. If $p=2$ this property is equivalent to the well-studied notion of unitarisability. We prove that amenable groups are $p$-isometrisable for all $p\in (1, \infty)$. Conversely, we show that every group containing a non-abelian free subgroup is not $p$-isometrisable for any $p\in (1, \infty)$. We also discuss some open questions and possible relations of $p$-isometrisability with the recently introduced Littlewood exponent $\mathrm{Lit}(\Gamma)$.

Keywords: unitarisability of groups, Dixmier problem, Banach spaces, $p$-spaces

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