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 groupH , 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.