## Algebras of functions on semitopological left-groups

HTML articles powered by AMS MathViewer

- by John F. Berglund and Paul Milnes PDF
- Trans. Amer. Math. Soc.
**222**(1976), 157-178 Request permission

## 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.

## References

- R. P. Hunter and L. W. Anderson,
*On the infinite subsemigroups of a compact semigroup*, Fund. Math.**74**(1972), no. 1, 1–19. MR**296204**, DOI 10.4064/fm-74-1-1-19 - John F. Berglund,
*On extending almost periodic functions*, Pacific J. Math.**33**(1970), 281–289. MR**412742**, DOI 10.2140/pjm.1970.33.281 - J. F. Berglund and K. H. Hofmann,
*Compact semitopological semigroups and weakly almost periodic functions*, Lecture Notes in Mathematics, No. 42, Springer-Verlag, Berlin-New York, 1967. MR**0223483**, DOI 10.1007/BFb0073920 - N. Bourbaki,
*Éléments de mathématique. Topologie générale. Chapitres 1 à 4*, Hermann, Paris, 1971. MR**0358652** - C. Corduneanu,
*Almost periodic functions*, Interscience Tracts in Pure and Applied Mathematics, No. 22, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. With the collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by Gitta Bernstein and Eugene Tomer. MR**0481915** - K. de Leeuw and I. Glicksberg,
*Applications of almost periodic compactifications*, Acta Math.**105**(1961), 63–97. MR**131784**, DOI 10.1007/BF02559535 - K. de Leeuw and I. Glicksberg,
*Almost periodic functions on semigroups*, Acta Math.**105**(1961), 99–140. MR**131785**, DOI 10.1007/BF02559536 - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Robert Ellis,
*Locally compact transformation groups*, Duke Math. J.**24**(1957), 119–125. MR**88674** - A. Grothendieck,
*Critères de compacité dans les espaces fonctionnels généraux*, Amer. J. Math.**74**(1952), 168–186 (French). MR**47313**, DOI 10.2307/2372076 - C. J. Knight, W. Moran, and J. S. Pym,
*The topologies of separate continuity. I*, Proc. Cambridge Philos. Soc.**68**(1970), 663–671. MR**267520**, DOI 10.1017/s0305004100076659 - Paul Milnes,
*Compactifications of semitopological semigroups*, J. Austral. Math. Soc.**15**(1973), 488–503. MR**0348030**, DOI 10.1017/S1446788700028858 - Theodore Mitchell,
*Topological semigroups and fixed points*, Illinois J. Math.**14**(1970), 630–641. MR**270356** - Vlastimil Pták,
*An extension theorem for separately continuous functions and its application to functional analysis*, Czechoslovak Math. J.**14(89)**(1964), 562–581 (English, with Russian summary). MR**172108**, DOI 10.21136/CMJ.1964.100640 - Chivukula Ramamohana Rao,
*Invariant means on spaces of continuous or measurable functions*, Trans. Amer. Math. Soc.**114**(1965), 187–196. MR**174938**, DOI 10.1090/S0002-9947-1965-0174938-0
E. C. Titchmarsh,

*The theory of functions*, 2nd ed., Oxford Univ. Press, London, 1939.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**222**(1976), 157-178 - MSC: Primary 43A60; Secondary 22A20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0422998-9
- MathSciNet review: 0422998