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. These ideas are applied to investigate to what extent the group algebra of integrable functions on an LCA group G , L 1 ( G ), characterizes the group.