An extension theorem for functions on semigroups

Milnes, Paul

doi:10.1090/S0002-9939-1976-0420153-5

An extension theorem for functions on semigroups
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Proc. Amer. Math. Soc. 55 (1976), 152-154 Request permission

Abstract:

For $S$ a semitopological semigroup, a continuous function on $S$ is said to be in $LMC(S)$ if its set of right translates is relatively compact in $C(S)$ for the topology of pointwise convergence on $S$. It is proved here that, if $S$ is a dense subsemigroup of a topological group $G$, then every function in $LMC(S)$ extends to a function continuous on $G$. This result generalizes earlier results that were arrived at independently by A. T. Lau and the present author. Some corollaries of this result are also presented.

References

John F. Berglund, On extending almost periodic functions, Pacific J. Math. 33 (1970), 281–289. MR 412742
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
R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Anthony To-ming Lau, Invariant means on dense subsemigroups of topolgical groups, Canadian J. Math. 23 (1971), 797–801. MR 288209, DOI 10.4153/CJM-1971-088-4
Paul Milnes, Extension of continuous functions on topological semigroups, Pacific J. Math. 58 (1975), no. 2, 553–562. MR 393332
Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356

Sur les espaces à structure uniforme et sur la topologie générale