Transmission of continuity to the Bohr topology

For a locally compact Abelian (LCA) group G, let G+ denote the group G endowed with its Bohr topology. With each piecewise affine map (defined below) α of G into another LCA group H, we show that there is associated a continuous map α+ of G+ into H+ which coincides with α on a dense open subset of G+. We study when α+ is a homeomorphism, provided that α has this property. ideas are applied to investigate to what extent the group algebra of integrable functions on an LCA group G, L1 (G), characterizes the group.