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Abstract harmonic analysis of generalised functions on locally compact semigroups with applications to invariant means

Published online by Cambridge University Press:  09 April 2009

James C. S. Wong
James C. S. Wong
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Canada, T2N IN4.
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Abstract

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Let S be a locally compact semigroup and M(S) its measure algebra. It is shown that the dual M(S)* is isometrically order isomorphic to the space GL(S) of all generalised functions on S first introduced by Šreǐder (1950). Moreover, convolutions of elements in each of the spaces M(S)* and GL(S) can be defined in such a way that the above isomorphism preserves convolutions. These results on representation of functionals in M(S)* by generalised functions practically open up a new chapter in abstract harmonic analysis. As an example, some applications to invariant means on locally compact semigroups are given.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 1 , February 1977 , pp. 84 - 94
Copyright
Copyright © Australian Mathematical Society 1977

References

Baker, A. C. and Baker, J. W. (1969), ‘Algebras of measures on a locally compact semigroup’, J. London Math. Soc. (2) 1 249–259.CrossRefGoogle Scholar
Baker, A. C. and Baker, J. W. (1970), ‘Algebras of measures on a locally compact semigroupII’, J. london Math. Soc. (2), 2, 651–659.CrossRefGoogle Scholar
Baker, A. C. and Baker, J. W. (1972), ‘Algebras of measures on a locally compact semigroup III’ J. London Math. Soc. (2) 4, 685–695.CrossRefGoogle Scholar
Glicksberg, I. (1961), ‘Weak compactness and separate continuity’, Pacific J. Math. 11, 205–214.CrossRefGoogle Scholar
Greenleaf, F. P. (1969), Invariant means on topological groups (Van Nostrand Math. Studies No. 16, Van Nostrand, New York).Google Scholar
Hart, G. (1970), Absolute continuous measures on semigroups (Ph.D. dissertation, Kansas State University, Kansas).Google Scholar
Hewitt, E. and Ross, K. A. (1963), Abstract harmonic analysis I (Springer-Verlag, Berlin).Google Scholar
Saka, K. (1974), ‘On a characterisation of some L-subalgebras in measure algebras’, J. London Math. Soc. (2) 9, 261–271.CrossRefGoogle Scholar
Šreider, Yu. A. (1950), ‘The structure of maximal ideals in rings of measures with convolution’, (Russian), Math. Sbornik (N.S.) 27 (69), 297–318. English translations (1953) in: Amer. Math. Soc. Transl. (First Series) No. 81, 365–391.Google Scholar
Williamson, J. H. (1967), ‘Harmonic analysis on semigroups’, J. London Math. Soc. 42, 1–41.CrossRefGoogle Scholar
Wong, J. C. S. (1969), ‘Topologically stationary locally compact groups and amenability’, Trans. Amer. Math. Soc. 144, 351–363.CrossRefGoogle Scholar
Wong, J. C. S. (1973), ‘An ergodic property of locally compact amenable semigroups’, Pacific J. Math. 48, 615–619.CrossRefGoogle Scholar
Wong, J. C. S. (1975), ‘Absolutely continuous measures on locally compact semigroups’, Canad. Math. Bull. 18 (1), 12–132.CrossRefGoogle Scholar