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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Countable groups of isometries on Banach spaces


Authors: Valentin Ferenczi and Elói Medina Galego
Journal: Trans. Amer. Math. Soc. 362 (2010), 4385-4431
MSC (2000): Primary 46B03, 46B04
Published electronically: March 12, 2010
MathSciNet review: 2608411
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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 \simeq \{-1,1\} \times H$, $ H$ finite and $ \dim X \geq \vert H\vert$, 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 (1986) 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 (1988) to prove that any complex Banach space of dimension at least $ 2$ may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least $ 4$ and with a complex structure 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.


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Additional Information

Valentin Ferenczi
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão, 1010 - Cidade Universitária, 05508-090 São Paulo, SP, Brazil
Email: ferenczi@ime.usp.br

Elói Medina Galego
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão, 1010 - Cidade Universitária, 05508-090 São Paulo, SP, Brazil
Email: eloi@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05034-8
PII: S 0002-9947(10)05034-8
Keywords: Group of isometries on Banach spaces, group representable in a Banach space, complex structures up to isometry.
Received by editor(s): June 13, 2007
Received by editor(s) in revised form: March 2, 2009
Published electronically: March 12, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.