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The centre of the second dual of a commutative semigroup algebra

Published online by Cambridge University Press:  24 October 2008

D. J. Parsons
D. J. Parsons
Affiliation:
Department of Pure Mathematics, University of Sheffield†
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If S is an infinite, discrete, commutative semigroup then the semigroup algebra l1(S) is a commutative Banach algebra. Its dual is l∞(S), which is isometrically iso-morphic to C(βS), the space of continuous functions on the Stone-Čech compactification of S. This fact enables us to identify the second dual of l1(S) with M(βS), the space of bounded regular Borel measures on βS. Endowed with the Arens product the second dual is also a Banach algebra, so it is natural to ask whether a product may be defined in M(βS) without reference to l1(S). In §4 this is shown to be possible even when S is a non-discrete semitopological semigroup, provided that the operation in S may be extended to make βS into a left-topological semigroup in the manner of, for example, [2] where further references may be found. (Note, however, that the construction there is of a right-topological semigroup.) Having done this we may use results on βS to provide information about the measure algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 1 , January 1984 , pp. 71 - 92
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1] Arens, R.. The adjoint of a bilinear operator. Proc. Amer. Math. Soc. 2 (1951), 839–848.CrossRefGoogle Scholar
[2] Baker, J. W. and Butcher, R. J.. The Stone-Čech compactification of a topological semi-group. Math. Proc. Cambridge Philos. Soc. 80 (1976), 102–107.CrossRefGoogle Scholar
[3] Bourbaki, N.. Intégration (Hermann 1952).Google Scholar
[4] Butcher, R. J.. Ph.D. Thesis, University of Sheffield, England, 1975.Google Scholar
[5] Cohen, D. E.. Groups of cohomological dimension one. Lecture Notes in Mathematics 245 (Springer-Verlag 1972).CrossRefGoogle Scholar
[6] Duncan, J. and Hosseiniun, S. A. R.. The second dual of a Banach algebra. Proc. Roy. Soc. Edinburgh, Sect. A. 84 (1979), 309–325.CrossRefGoogle Scholar
[7] Gillman, L. and Jerison, M.. Rings of continuous functions (Van Nostrand 1963).Google Scholar
[8] Glicksberg, I.. Weak compactness and separate continuity. Pac. J. Math. 11 (1961), 205–214.CrossRefGoogle Scholar
[9] Hindman, N.. Minimal ideals and cancellation in βN. Semigroup Forum 25 (1982), 291–310.CrossRefGoogle Scholar
[10] Holt, D. F.. Uncountable locally finite groups have only one end. Bull. Lond. Math. Soc. 13 (1981), 557–560.CrossRefGoogle Scholar
[11] Hopf, H.. Enden offener Räume und unendenliche diskontinuerliche Gruppen. Comm. Math. Helv. 16 (1943), 81–100.CrossRefGoogle Scholar
[12] Knowles, J. D.. Measures on topological spaces. Proc. Lond. Math. Soc. (3), 17 (1967), 139–156.CrossRefGoogle Scholar
[13] Parsons, D. J.. Ph.D. Thesis, University of Sheffield, England, 1983.Google Scholar
[14] Pym, J. S. and Vasudeva, H. L.. Semigroup structure in compactifications of ordered semigroups. Czech. Math. J. 27 (102) (1977), 528–544.CrossRefGoogle Scholar
[15] Rotman, J. J.. The theory of groups: an introduction (Allyn & Bacon 1965).Google Scholar
[16] Willard, S. J.. General topology (Addison-Wesley 1970).Google Scholar