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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Abstract functions with continuous differences and Namioka spaces

Authors: Bolis Basit and Hans Günzler
Journal: Trans. Amer. Math. Soc. 348 (1996), 4489-4500
MSC (1991): Primary 28B05, 39A05; Secondary 90D05, 54C05, 54E35
MathSciNet review: 1373629
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Abstract: Let $G$ be a semigroup and a topological space. Let $X$ be an Abelian topological group. The right differences $\triangle _{h} \varphi $ of a function $\varphi : G \to X$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\triangle _{h} \varphi $ be continuous at the identity $e$ of $G$ for all $h$ in a neighbourhood $U$ of $e$. We give conditions on $X$ or range $\varphi $ under which $\varphi $ is continuous for any topological space $G$. We also seek conditions on $G$ under which we conclude that $\varphi $ is continuous at $e$ for arbitrary $X$. This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

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  • 1. L. Amerio and G. Prouse, Almost Periodic Functions and Functional Equations, Van-Nostrand Reinhold Company, 1971. MR 48:419
  • 2. B. Basit and M. Emam, Differences of functions in locally convex spaces and applications to almost periodic and almost automorphic functions, Annales Polonici Math. 41 (1983), 193-201. MR 85d:43005
  • 3. B. Basit and A. J. Pryde, Differences of vector-valued functions on topological groups, Proc. Amer. Math. Soc. 124 (1996), 1969--1975. CMP 95:08
  • 4. J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups: Function Spaces, Compactification, Representations, Canadian Mathematical Society Series of Monographs, A Wiley-Interscience Publication, 1989. MR 91b:43001
  • 5. C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 22:5872
  • 6. J. Calbrix et J. P. Troallic, Application séparément continues, C.R.Acad.Sci. Paris, série A-B, 288 (1979), 647-648. MR 80c:54009
  • 7. G. Choquet, Lectures on Analysis, Vol. 1, Benjamin, New York and Amesterdam, 1969. MR 40:3252
  • 8. J.P.R Christensen, Joint continuity of separately continuous functions, Proc. Amer. Math. Soc. 82 (1981), 455-461. MR 82h:54012
  • 9. C. Datry and G. Muraz, Analyse harmonique dans les modules de Banach I : propriétés générales, Bull. Science Mathematique 119 (1995), 299-337. CMP 95:15
  • 10. F. Galvin, G. Muraz et P. Szeptycki, Fonctions aux différences $f(x)-f(a+x)$ continues, C.R.Acad.Sci. Paris, série I, 315 (1992), 397-400. MR 94b:39035
  • 11. H. Günzler, Integration of almost periodic functions, Math. Zeit. 102, (1967), 253- 287. MR 36:3066
  • 12. N. J. Kalton, Subseries convergence in topological groups and vector spaces, Israel J. Math. 10 (1970), 402-412. MR 45:3628
  • 13. W. Maak, Fastperiodische Funktionen, Springer-Verlag, 1967. MR 35:5860
  • 14. I. Namioka, Separate continuity and joint continuity, Pacific Journal of Math. 51 (1974), 515-531. MR 51:6693
  • 15. J. Saint-Raymond, Jeux topologiques et espaces de Namioka, Proc. Amer. Math. Soc. 87 (1983), 499-504. MR 83m:54060
  • 16. F. Watbled, Ensemble de Rosenthal pour des Fonctions a Valeurs Banach, C.R.Acad.Sci. Paris, série I, 318 (1994), 333-336. MR 94m:43007

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Additional Information

Bolis Basit
Affiliation: Department of Mathematics, Monash University, Clayton Vic. 3168, Australia

Hans Günzler
Affiliation: Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str., 424098 Kiel, Deutschland

Keywords: Differences, weak continuity, Namioka spaces, right uniform continuity, Baire spaces, Banach spaces not containing $c_{0}$
Received by editor(s): May 4, 1995
Article copyright: © Copyright 1996 American Mathematical Society