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Transactions of the American Mathematical Society

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



Algebras of functions on semitopological left-groups

Authors: John F. Berglund and Paul Milnes
Journal: Trans. Amer. Math. Soc. 222 (1976), 157-178
MSC: Primary 43A60; Secondary 22A20
MathSciNet review: 0422998
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Abstract: We consider various algebras of functions on a semitopological left-group $ S = X \times G$, the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products $ S \times T$ of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.

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  • [1] R. P. Hunter and L. W. Anderson, On the infinite sub-semigroups of a compact semigroup, Fund. Math. 74 (1972), 1-19. MR 45 #5265. MR 0296204 (45:5265)
  • [2] J. F. Berglund, On extending almost periodic functions, Pacific J. Math. 33 (1970), 281-289. MR 0412742 (54:863)
  • [3] J. F. Berglund and K. H. Hofmann, Compact semitopological semigroups and weakly almost periodic functions, Lecture Notes in Math., no. 42, Springer-Verlag, Berlin, 1967. MR 0223483 (36:6531)
  • [4] N. Bourbaki, Topologie générale, Hermann, Paris, 1971. MR 0358652 (50:11111)
  • [5] C. Corduneanu, Almost periodic functions, Interscience, New York, 1968. MR 0481915 (58:2006)
  • [6] K. deLeeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63-97. MR 24 #A1632. MR 0131784 (24:A1632)
  • [7] -, Almost periodic functions on semigroups, Acta Math. 105 (1961), 99-140. MR 24 #A1633. MR 0131785 (24:A1633)
  • [8] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [9] R. Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), 119-125. MR 19, 561. MR 0088674 (19:561b)
  • [10] A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186. MR 13, 857. MR 0047313 (13:857e)
  • [11] C. J. Knight, W. Moran and J. S. Pym, The topologies of separate continuity. I, Proc. Cambridge Philos. Soc. 68 (1970), 663-671. MR 42 #2422. MR 0267520 (42:2422)
  • [12] P. Milnes, Compactifications of semitopological semigroups, J. Austral. Math. Soc. 15 (1973), 448-503. MR 50 #528. MR 0348030 (50:528)
  • [13] T. Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630-641. MR 42 #5245. MR 0270356 (42:5245)
  • [14] V. Pták, An extension theorem for separately continuous functions and its application to functional analysis, Czechoslovak Math. J. 14 (89) (1964), 562-581. MR 30 #2334. MR 0172108 (30:2334)
  • [15] C. R. Rao, Invariant means on spaces of continuous or measurable functions, Trans. Amer. Math. Soc. 114 (1965), 187-196. MR 30 #5128. MR 0174938 (30:5128)
  • [16] E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, London, 1939.

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Keywords: Semitopological semigroup, left-group, weakly almost periodic function, compactification, analytic function
Article copyright: © Copyright 1976 American Mathematical Society

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