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Primitive idempotent measures on compact semitopological semigroups

Published online by Cambridge University Press:  09 April 2009

Stephen T. L. Choy
Stephen T. L. Choy
Affiliation:
University of Hong Kong and University of Singapore
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For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 4 , June 1972 , pp. 451 - 455
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Berglund, J. F. and Hofmann, K. H., Compact semitopological semigroups and weakly almost periodic functions, Lecture Notes in Mathematics 42 (Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
[2]Choy, S. T. L., ‘Idempotent measures on compact semigroups’, Proc. London Math. Soc. (to appear).Google Scholar
[3]Collins, H. S., Primitive idempotents in the semigroup of measures', Duke Math. J. 27 (1960), 397–400.CrossRefGoogle Scholar
[4]Collins, H. S., ‘The kernel of a semigroup of measures’, Duke Math. J. 28 (1961), 387–391.CrossRefGoogle Scholar
[5]Duncan, J., ‘Primitive idempotent measures on compact semigroups’, Proc. Edinburgh Math. Soc. (to appear).Google Scholar
[6]Glicksberg, I., ‘Weak compactness and separate continuity’, Pacific J. Math. 11 (1961), 205–214.CrossRefGoogle Scholar
[7]Hewitt, E. and Ross, K. A., Abstract harmonic analysis (Springer-Verlag, 1963).Google Scholar
[8]Pym, J. S., ‘Idempotent measures on semigroups’, Pacific J. Math. 12 (1962), 685–698.CrossRefGoogle Scholar
[9]Pym, J. S., ‘Weakly separately continuous measure algebras’, Math. Ann. 175 (1968), 207–219.CrossRefGoogle Scholar
[10]Pym, J. S., ‘Idempotent probability measures on compact semi-topological semigroups’, Proc. American Math. Soc. 21 (1969), 499–501.Google Scholar