Relations between Locally Compact Abelian Groups with Isomorphic Group Algebras

Let G be a locally compact Abelian group and consider the group algebra L1(G) of all complex-valued absolutely integrable functions on G. Some invariants are found that are necessarily shared by groups with isomorphic group algebras, such as dimension, ranks or minimal divisible extensions in the case of torsion-free groups. This sheds some light on the question of finding out to what extent the algebra structure of L1(G) reflects the topological group structure of G. It is shown in particular that some classes of torsion-free locally compact Abelian groups are characterized by their group algebra.