The set of idempotents in the weakly almost periodic compactification of the integers is not closed

Bordbar, B.; Pym, J.

doi:10.1090/S0002-9947-99-02273-4

The set of idempotents in the weakly almost periodic compactification of the integers is not closed
HTML articles powered by AMS MathViewer

by B. Bordbar and J. Pym PDF

Trans. Amer. Math. Soc. 352 (2000), 823-842 Request permission

Abstract:

This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of $(\mathbb {N}, +)$ and $(\mathbb {Z} ,+)$ are closed.

References

John F. Berglund, Problems about semitopological semigroups, Semigroup Forum 19 (1980), no. 4, 373–383. MR 569815, DOI 10.1007/BF02572530
B Bordbar, Weakly almost periodic functions on $\mathbb {N}$ with a negative base, J London Math Soc (to appear).
B Bordbar and J Pym, The weakly almost periodic compactification of a direct sum of finite groups, Math Proc Cambridge Phil Soc 124 (1998), 421–449.
G. Brown and W. Moran, The idempotent semigroups of compact monothetic semigroups, Proc. Roy. Irish Acad. Sect. A 72 (1972), 17–33. MR 304547
Janusz J. Charatonik, Contractibility of curves, Matematiche (Catania) 46 (1991), no. 2, 559–592 (1993). MR 1216780
K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. MR 131784, DOI 10.1007/BF02559535
Gerald A. Edgar, Measure, topology, and fractal geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. MR 1065392, DOI 10.1007/978-1-4757-4134-6
Neil Hindman, The ideal structure of the space of $\kappa$-uniform ultrafilters on a discrete semigroup, Rocky Mountain J. Math. 16 (1986), no. 4, 685–701. MR 871030, DOI 10.1216/RMJ-1986-16-4-685
N Hindman and D Strauss, Algebra in the Stone-Čech Compactification, de Gruyter, Berlin, 1998.
Talin Papazyan, Oids, finite sums and the structure of the Stone-Čech compactification of a discrete semigroup, Semigroup Forum 42 (1991), no. 3, 265–277. MR 1092958, DOI 10.1007/BF02573424
John Pym, Semigroup structure in Stone-Čech compactifications, J. London Math. Soc. (2) 36 (1987), no. 3, 421–428. MR 918634, DOI 10.1112/jlms/s2-36.3.421
Wolfgang Ruppert, Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Mathematics, vol. 1079, Springer-Verlag, Berlin, 1984. MR 762985, DOI 10.1007/BFb0073675
W. A. F. Ruppert, Compact semitopological semigroups, The analytical and topological theory of semigroups, De Gruyter Exp. Math., vol. 1, de Gruyter, Berlin, 1990, pp. 133–170. MR 1072787
W. A. F. Ruppert, On signed $\textbf {a}$-adic expansions and weakly almost periodic functions, Proc. London Math. Soc. (3) 63 (1991), no. 3, 620–656. MR 1127152, DOI 10.1112/plms/s3-63.3.620
T. T. West, Weakly compact monothetic semigroups of operators in Banach spaces, Proc. Roy. Irish Acad. Sect. A 67 (1968), 27–37 (1968). MR 239467