A note on generalized characters

Abstract

For a compactly generated LCA group G, it is shown that the set H(G) of all generalized characters on G equipped with the compact-open topology is a LCA group and H(G) = G^{^} (the dual group of G) if and only if G is compact. Both results fail for arbitrary LCA groups. Further, if G is second countable, then the Gel'fand space of the commutative convolution algebra C_c (G) equipped with the inductive limit topology is topologically homeomorphic to H(G).