Topological groups whose underlying spaces are separable Fréchet manifolds

Abstract

Let G be a topological group and let G* denote the space of all Lebesgue measurable functions from the unit interval [0, 1] into G with the topology of convergence in measure. With this topology and with pointwise multiplication as the group operation, G* is a topological group. If G is separable and has a complete metric and has more than one point, then Bessaga and Peiczyήski have shown that G* is homeomorphic to h, separable infinite-dimensional Hubert space. This fact is used in this paper to show the existence of separable Frechet manifolds which are topological groups and which have certain algebraic and topological properties.