Countable groups of isometries on Banach spaces

Abstract

A group G G is representable in a Banach space X X if G G is isomorphic to the group of isometries on X X in some equivalent norm. We prove that a countable group G G is representable in a separable real Banach space X X in several general cases, including when G ≃ { − 1 , 1 } × H G \simeq \{-1,1\} \times H , H H finite and dim ⁡ X ≥ | H | \dim X \geq |H| , or when G G contains a normal subgroup with two elements and X X is of the form c 0 ( Y ) c_0(Y) or ℓ p ( Y ) \ell _p(Y) , 1 ≤ p > + ∞ 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 X and a countable bounded group G G of isomorphisms on X X containing − I d -Id , there exists an equivalent norm on X X for which G G is equal to the group of isometries on X X . We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 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 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.