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Abstract: A group G is representable in a Banach space X if G is isomorphic to thegroup of isometries on X in some equivalent norm. We prove that a countablegroup G is representable in a separable real Banach space X in several generalcases, including when $G=\{-1,1\} \times H$, H finite and $\dim X \geq |H|$, orwhen G contains a normal subgroup with two elements and X is of the form c 0Yor $\ell pY$, $1 \leq p <+\infty$. This is a consequence of a result inspiredby methods of S. Bellenot and stating that under rather general conditions on aseparable real Banach space X and a countable bounded group G of isomorphismson X containing -Id, there exists an equivalent norm on X for which G is equalto the group of isometries on X.We also extend methods of K. Jarosz to prove that any complex Banach space ofdimension at least 2 may be renormed to admit only trivial real isometries, andthat any real Banach space which is a cartesian square may be renormed to admitonly trivial and conjugation real isometries. It follows that every real spaceof dimension at least 4 and with a complex structure up to isomorphism may berenormed to admit exactly two complex structures up to isometry, and that everyreal cartesian square may be renormed to admit a unique complex structure up toisometry.



Author: Valentin Ferenczi, Eloi Medina Galego

Source: https://arxiv.org/







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