An application of the Bohr topology to group algebra isomorphisms
by
Jorge Galindo
Depto. de Matematicas, Univ. Jaume I, Castellon, Spain

It is well known that every continuous homomorphism between two MAP Abelian groups is also continuous when G and H are endowed with their respective Bohr topologies.

A first attempt to extend this result to a wider class of mappings leads to considering affine and piecewise affine mappings. On the other hand, it is a theorem of P. J. Cohen that the group algebras of two LCA groups are isomorphic if and only if there exists a piecewise affine homeomorphism between their dual groups.

Thus, the knoweldge of piecewise affine mappings concerns both the study of Bohr topologies and the investigation about the following question posed by Rudin: Suppose the algebras L¹(G₁) and L¹(G₂) are isomorphic. What can be said about the relation between G₁ and G₂?.

In this talk we will show some results on transmission of continuity for piecewise affine mappings with applications to the question above. It will be considered first the case of a piecewise affine mapping. Eventhough continuous piecewise affine mappings need not be continuous for the Bohr topologies, we show that with each piecewise affine map \alpha of a LCA group G into another LCA group H, there is associated a continuous map \alpha⁺ of G⁺ into H⁺ which coincides with \alpha on a dense open subset of G⁺. This considerations become more interesting when one regards the case of a piecewise affine homeomorphism \alpha. Again, the associated map \alpha⁺ need not be a homeomorphism in general but, in some special cases, it can be shown that it actually is. These results help to find new ties between LCA groups whose group algebras are isomorphic. For instance, the minimal divisible extensions of two torsion free LCA groups with isomorphic group algebras are topologically isomorphic. Without restricting the range of groups under consideration, such a general result is impossible to prove, since, as it will be shown in this talk, a wide family of non-isomorphic groups with isomorphic group algebras can be found. However, it is possible to prove that two LCA groups with isomorphic group algebras, necessarily have the same dimension.

Date received: February 24, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caam-14.