Countable groups of isometries on Banach spaces

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 $\dim X \geq |H|$, 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 (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.