Abstract functions with continuous differences and Namioka spaces
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- by Bolis Basit and Hans Günzler PDF
- Trans. Amer. Math. Soc. 348 (1996), 4489-4500 Request permission
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.References
- Gheorghe Pic, On a theorem of J. G. Thompson, Rev. Roumaine Math. Pures Appl. 17 (1972), 1419–1422. MR 322055
- Bolis Basit and Magdy Emam, Differences of functions in locally convex spaces and applications to almost periodic and almost automorphic functions, Ann. Polon. Math. 41 (1983), no. 3, 193–201. MR 730302, DOI 10.4064/ap-41-3-193-201
- B. Basit and A. J. Pryde, Differences of vector-valued functions on topological groups, Proc. Amer. Math. Soc. 124 (1996), 1969–1975.
- John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR 999922
- C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
- Jean Calbrix and Jean-Pierre Troallic, Applications séparément continues, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 13, A647–A648 (French, with English summary). MR 534521
- Gustave Choquet, Lectures on analysis. Vol. I: Integration and topological vector spaces, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Edited by J. Marsden, T. Lance and S. Gelbart. MR 0250011
- Jens Peter Reus Christensen, Joint continuity of separately continuous functions, Proc. Amer. Math. Soc. 82 (1981), no. 3, 455–461. MR 612739, DOI 10.1090/S0002-9939-1981-0612739-1
- C. Datry and G. Muraz, Analyse harmonique dans les modules de Banach I : propriétés générales, Bull. Science Mathematique 119 (1995), 299-337.
- Fred Galvin, Gilbert Muraz, and Pawel Szeptycki, Fonctions aux différences $f(x)-f(a+x)$ continues, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 4, 397–400 (French, with English and French summaries). MR 1179045
- Hans Günzler, Integration of almost periodic functions, Math. Z. 102 (1967), 253–287. MR 219997, DOI 10.1007/BF01110910
- N. J. Kalton, Subseries convergence in topological groups and vector spaces, Israel J. Math. 10 (1971), 402–412. MR 294558, DOI 10.1007/BF02771728
- W. Maak, Fastperiodische Funktionen, Die Grundlehren der mathematischen Wissenschaften, Band 61, Springer-Verlag, Berlin-New York, 1967 (German). Zweite, korrigierte Auflage. MR 0215015, DOI 10.1007/978-3-642-86687-6
- I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515–531. MR 370466, DOI 10.2140/pjm.1974.51.515
- Jean Saint-Raymond, Jeux topologiques et espaces de Namioka, Proc. Amer. Math. Soc. 87 (1983), no. 3, 499–504 (French, with English summary). MR 684646, DOI 10.1090/S0002-9939-1983-0684646-1
- Frédérique Watbled, Ensembles de Rosenthal pour des fonctions à valeurs Banach, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 4, 333–336 (French, with French summary). MR 1267610
Additional Information
- Bolis Basit
- Affiliation: Department of Mathematics, Monash University, Clayton Vic. 3168, Australia
- Email: bbasit@vaxc.cc.monash.edu.au
- Hans Günzler
- Affiliation: Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str., 424098 Kiel, Deutschland
- Email: guenzler@math.uni-kiel.de
- Received by editor(s): May 4, 1995
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 4489-4500
- MSC (1991): Primary 28B05, 39A05; Secondary 90D05, 54C05, 54E35
- DOI: https://doi.org/10.1090/S0002-9947-96-01715-1
- MathSciNet review: 1373629