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Relations between Locally Compact Abelian Groups with Isomorphic Group Algebras
Author(s) -
Galindo Jorge
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079900811x
Subject(s) - mathematics , abelian group , rank of an abelian group , non abelian group , g module , elementary abelian group , group algebra , pure mathematics , free abelian group , torsion (gastropod) , group (periodic table) , algebra over a field , topological group , locally compact group , metabelian group , locally compact space , combinatorics , topology (electrical circuits) , physics , quantum mechanics , medicine , surgery
Let G be a locally compact Abelian group and consider the group algebra L 1 ( 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 L 1 ( 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.