  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.
