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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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  • 1. W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Springer Lecture Notes in Mathematics 1466, 1991. MR 93b:46005
  • 2. W.W. Comfort, F.J. Trigos-Arrieta, Remarks on a theorem of Glicksberg, pp. 25-33 in: General topology and applications, Proceedings of a conference in Staten Island, 1989, (Eds.: S.J. Andima, R. Kopperman, P.R. Misra, J.Z. Reichman, A.R. Todd), Dekker, New York, 1991. MR 92k:54042
  • 3. S. Dierolf, S. Warken, Some examples in connection with Pontryagin's duality theorem, Arch. Math. 30 (1978), 599-605. MR 58:16943
  • 4. I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269-276. MR 27:5856
  • 5. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis I, 2nd ed., Springer, 1979. MR 81k:43001
  • 6. S. Kaplan, Extensions of the Pontrjagin duality I: Infinite products, Duke Math. J. 15 (1948), 649-658. MR 10:233c
  • 7. E. Martín-Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 (1995), 3563-3566. MR 96a:22002
  • 8. S.A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge University Press, Cambridge 1977. MR 56:529
  • 9. N. Noble, $k$-groups and duality, Trans. Amer. Math. Soc. 151 (1970), 551-561. MR 42:4963
  • 10. V. Pestov, Free abelian topological groups and the Pontryagin - van Kampen duality, Bull. Austral. Math. Soc. 52 (1995), 297-311. MR 96k:22002
  • 11. D. Remus, Topological groups without non-trivial characters, pp. 477-484, in: General Topology and its Relations to Modern Analysis and Algebra VI, Proc. Sixth Prague Topological Symposium 1986, (Ed.: Z. Frolik), Heldermann Verlag, Berlin, 1988. MR 89f:22002
  • 12. M.F. Smith, The Pontrjagin duality theorem in linear spaces, Ann. of Math (2) 56 (1952), 248-253. MR 14:183a

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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

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