Countable groups of isometries on Banach spaces
| dc.creator | Ferenczi, Valentin | |
| dc.creator | Galego, Eloi Medina | |
| dc.date | 2007-06-26 | |
| dc.date | 2007-07-30 | |
| dc.date.accessioned | 2026-07-07T08:20:39Z | |
| dc.date.available | 2026-07-07T08:20:39Z | |
| dc.description | 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. | |
| dc.description | 44 pages | |
| dc.identifier | https://arxiv.org/abs/0706.3861 | |
| dc.identifier | http://arxiv.org/abs/0706.3861 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/135130 | |
| dc.subject | Functional Analysis | |
| dc.subject | 46B03, 46B04 | |
| dc.title | Countable groups of isometries on Banach spaces | |
| dc.type | text |