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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

$ ({\rm CA})$ topological groups


Author: David Zerling
Journal: Proc. Amer. Math. Soc. 54 (1976), 345-351
MSC: Primary 22D05
MathSciNet review: 0412337
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Abstract: A locally compact topological group $ G$ is called $ (CA)$ if the group of inner automorphisms of $ G$ is closed in the group of all bicontinuous automorphisms of $ G$. We show that each non-$ (CA)$ locally compact connected group $ G$ can be written as a semidirect product of a $ (C\,A)$ locally compact connected group by a vector group. This decomposition yields a natural dense imbedding of $ G$ into a $ (C\,A)$ locally compact connected group $ P$, such that each bicontinuous automorphism of $ G$ can be extended to a bicontinuous automorphism of $ P$. This imbedding and extension property enables us to derive a sufficient condition for the normal part of a semidirect product decomposition of a $ (C\,A)$ locally compact connected group to be $ (C\,A)$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0412337-7
Article copyright: © Copyright 1976 American Mathematical Society