Countable groups of isometries on Banach spaces

Abstract

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} × H, H finite and dim X ≥ |H|, or when G contains a normal subgroup with two elements and X is of the form c 0 (Y ) or p (Y ), 1 ≤ p < +∞. 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.

Complex structures on twisted Hilbert spaces

“…At the other extreme, we find the classical spaces ℓ p or L p , which admit a unique complex structure up to C-linear isomorphism. A simple proof of this fact was provided by N.J. Kalton and appears in [28,Thm. 22].…”

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

“…In [9], Ferenczi and Rosendal generalized results of [7] to certain uncountable Polish groups, and also defined the concept of distinguished family for the action of a group G on a Banach space X. It is clear that if G is an isometry group with a distinguished point, G is SOT-discrete.…”

Light groups of isometries and polyhedrality of Banach spaces

“…Since ϕ α n+1 is an increasing homeomorphism and y n, Ferenczi and Rosendal generalized results of [FG10] in [FR11] to certain uncountable Polish groups and also defined the concept of distinguished family for X in relation to G as a finite subset…”

“…In Chapter 4 we will connect the concepts of light groups with the concept of distinguished point, defined by Ferenczi and Rosendal in [FR11]. Ferenczi and Galego investigated in [FG10] groups that may be seen as the group of isometries of a Banach space under some renorming. Among other results, they prove that if X is a separable Banach space with LUR norm • and if G is an infinite countable isometry group of X such that − Id ∈ G and such that G admits a point x ∈ X with inf g =Id gx − x > 0, then G = Isom(X, ||| • |||) for some equivalent norm ||| • ||| on X.…”