Displaying Polish groups on separable Banach spaces

Abstract

A display of a topological group G on a Banach space X is a topological isomor- phism of G with the isometry group Isom(X, ⦀·⦀) for some equivalent norm ⦀ ⦀on X, where the latter group is equipped with the strong operator topology. Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the in_nite symmetric group S∞ containing a non-trivial central involution admits a display on any of the classical spaces c₀, C([0, 1]), ℓp and Lp for 1 ≤ p < ∞. Also, for any Polish group G, there exists a separable space X on which {-1, 1} × G has a display.