Locally Bicompact Abelian Groups and Their Character Groups

Abstract

The theory of character groups of Abelian groups, long at a standstill near the beginning, has been taken up recently by L. Pontrjagin.1 He showed how the well known simple isomorphism of a finite Abelian group and its character group is a very special case of a dual one-to-one correspondence between compact separable Abelian groups and discrete countable Abelian groups; each group is the character group of its dual group and the finite groups are selfcorresponding. The proof of this extension uses a certain amount of direct group theoretic-topological reasoning, and an existence theorem of characters of compact separable Abelian groups. This existence theorem is the result of a long analysis, that can be founded either on the Haar measure in locally compact groups2 or on von Neumann's theory of almost periodic functions in groups3 as will be done in this paper. Von Neumann remarked that the whole theory can be extended without any additional effort to bicompact and (unrestricted) discrete groups. The explicit relation between the discrete groups and its bicompact character group can be found in a paper by J. W. Alexander.4 Using his group theoretic duality theorem as a basis, L. Pontrjagin5 then proved that each separable locally compact, connected Abelian group is the direct sum of a finite dimensional translation group and a compact group. A consideration of his proofs shows that the only result of the theory of character groups of compact groups used, is the lemma that each compact separable group contains in an arbitrary neighborhood of the identity element a closed subgroup with a particularly simple factor group (compare III, 6). A suitable modification of Pontrjagin's result could be extended to arbitrary separable locally compact Abelian groups5a and then, according to von Neumann's remark mentioned above, to all locally bicompact Abelian groups.