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On the continuity of the evaluation mapping associated with a group and its character group

Author: Gerhard Turnwald
Journal: Proc. Amer. Math. Soc. 126 (1998), 3413-3415
MSC (1991): Primary 22A05; Secondary 22D35
MathSciNet review: 1452831
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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.

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Additional Information

Gerhard Turnwald
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Keywords: Character group, reflexive group, evaluation mapping
Received by editor(s): October 30, 1996
Received by editor(s) in revised form: March 21, 1997
Communicated by: Roe Goodman
Article copyright: © Copyright 1998 American Mathematical Society