This is an announcement for the paper "Countable groups of isometries on Banach spaces" by Valentin Ferenczi and Eloi Medina Galego.

Abstract: A group $G$ is representable in a Banach space $X$ if $G$ is isomorphic to the group of isometries on $X$ in some equivalent norm. We prove that a countable group $G$ is representable in a separable real Banach space $X$ in several general cases, including when $G={-1,1} \times H$, $H$ finite and $\dim X \geq |H|$, or when $G$ contains a normal subgroup with two elements and $X$ is of the form $c_0(Y)$ or $\ell_p(Y)$, $1 \leq p <+\infty$. This is a consequence of a result inspired by methods of S. Bellenot and stating that under rather general conditions on a separable real Banach space $X$ and a countable bounded group $G$ of isomorphisms on $X$ containing $-Id$, there exists an equivalent norm on $X$ for which $G$ is equal to the group of isometries on $X$. We also extend methods of K. Jarosz to prove that any complex Banach space of dimension at least $2$ may be renormed to admit only trivial real isometries, and that any real Banach space which is a cartesian square may be renormed to admit only trivial and conjugation real isometries. It follows that every real space of dimension at least $4$ and with a complex structure up to isomorphism may be renormed to admit exactly two complex structures up to isometry, and that every real cartesian square may be renormed to admit a unique complex structure up to isometry.

Archive classification: math.FA

Mathematics Subject Classification: 46B03; 46B04

Remarks: 43 pages

The source file(s), ferenczigalego_isometries.tex: 104441 bytes, is(are) stored in gzipped form as 0706.3861.gz with size 29kb. The corresponding postcript file has gzipped size 137kb.

Submitted from: ferenczi@ccr.jussieu.fr

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