Countable groups of isometries on Banach spaces

TL;DR

This paper investigates conditions under which countable groups can be realized as isometry groups of Banach spaces, introduces new renorming techniques, and explores the uniqueness of complex structures on real Banach spaces.

Contribution

It provides new methods for representing countable groups as isometry groups of Banach spaces and extends techniques to analyze the uniqueness of complex structures.

Findings

Countable groups can be represented as isometry groups under various conditions.

Banach spaces can be renormed to restrict or specify their isometry groups.

Unique complex structures can be achieved through renorming in certain real 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 $G=\{-1,1\} \times H$, 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.