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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the continuity of the evaluation mapping associated with a group and its character group
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by Gerhard Turnwald PDF
Proc. Amer. Math. Soc. 126 (1998), 3413-3415 Request permission

Abstract:

For an abelian Hausdorff group $G$, let $G^{\ast }$ denote the character group endowed with the compact-open topology and let $\alpha _{G}:G\rightarrow G^{\ast \ast }$ denote the canonical homomorphism. We show that the evaluation mapping from $G^{\ast }\times G$ into the torus is continuous if and only if $G^{\ast }$ is locally compact and $\alpha _{G}$ is continuous. If $\alpha _{G}$ is injective and open, then the evaluation mapping is continuous if and only if $G$ is locally compact. Several examples and counterexamples are given.
References
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Additional Information
  • Gerhard Turnwald
  • Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
  • Email: gerhard.turnwald@uni-tuebingen.de
  • Received by editor(s): October 30, 1996
  • Received by editor(s) in revised form: March 21, 1997
  • Communicated by: Roe Goodman
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3413-3415
  • MSC (1991): Primary 22A05; Secondary 22D35
  • DOI: https://doi.org/10.1090/S0002-9939-98-04412-8
  • MathSciNet review: 1452831