Displaying Polish groups on separable Banach spaces (Q2906823)

It is proved that each closed subgroup \(G\) of the symmetric group \(S_\infty\) with a non-trivial central involution is topologically isomorphic to the isometry group \(\mathrm{Iso}(X)\) of a Banach space \(X\) which is isomorphic to any of the classical Banach spaces: \(c_0\), \(C([0,1])\), \(\ell_p\), \(L_p\) for \(1\leq p<\infty\). Another main result says that for every Polish group \(G\) the group \(\{-1,1\}\times G\) is topologically isomorphic to the isometry group of some separable Banach space \(X\). The paper ends with some open problems. One of them asks if every Polish group with a non-trivial central involution is topologically isomorphic to the isometry group of some separable Banach spaces.