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Bochner's Theorem and the Hausdorff Moment Theorem on foundation Topological Semigroups

Published online by Cambridge University Press:  20 November 2018

M. Lashkarizadeh Bami
M. Lashkarizadeh Bami*
Affiliation:
University of Isfahan, Isfahan, Iran
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One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 785 - 809
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Baker, A. C. and Baker, J. W., Duality of topological semigroups with involutions, J. London Math. Soc. 44 (1969), 251–260.Google Scholar
2. Baker, A. C. and Baker, J. W., Algebras of measures on a locally compact semigroup, J. London Math. Soc. (2) (1969), 249–259.Google Scholar
3. Baker, A. C. and Baker, J. W., Algebras of measures on a locally compact semigroup II, J. London Math. Soc. (2) (1970), 651–659.Google Scholar
4. Baker, A. C. and Baker, J. W., Algebras of measures on a locally compact semigroup III, J. London Math. Soc. (2) 4 (1972), 685–695.Google Scholar
5. Berg, C., Christensen, P. R. and Resel, P., Positive-definite functions on abelian semigroups, Maths. Ann. 223 (1976), 253–272.Google Scholar
6. Berg, C. and Maserick, P. H., Exponentially bounded positive-definite functions, preprint.CrossRefGoogle Scholar
7. Dunkl, C. F. and Ramirez, D. E., Representations of commutative semitopological semigroups, Lecture Notes in Mathematics 435 (Springer-Verlag, Heidelberg, 1975).Google Scholar
8. Gelfand, I. M., Raikov, D. A. and Shilov, G. E., Commutative normed rings (Chelsea, New York, 1964).Google Scholar
9. Hewitt, E. and Ross, K. A., Abstract harmonic analysis I (Springer-Verlag, Heidelberg, 1970).Google Scholar
10. Hewitt, E. and Ross, K. A., Abstract harmonic analysis II (Springer-Verlag, Heidelberg, 1970).Google Scholar
11. Kelley, J. L., General topology (Prinston University Press, New Jersey, 1955).Google Scholar
12. Bami, M. Lashkarizadeh, Representations of foundation semigroups and their algebras, Can. J. Math. 37 (1985), 29–47.Google Scholar
13. Lindhal, R. J. and Maserick, P. H., Positive-definite functions on involution semigroups, Duke Math. J. (1971), 771–782.Google Scholar
14. Maserick, P. H., Spectral theory of operator-valued transformations, J. of Math. Anal, and Appl. 41 (1973), 497–507.Google Scholar
15. Phelps, R. P., Lectures on Choquefs theorem (Princeton, 1960).Google Scholar
16. Rudin, W., Fourier analysis on groups (Interscience, New York, 1952).Google Scholar
17. Sleijpen, G. L. G., Convolution measures algebras on semigroups (thesis, Catholic University, Toernooiveld, Nijmegen, 1976).Google Scholar
18. Sleijpen, G. L. G., Norm-and weak-continuity of translations of measures; a counterexample, Semigroup Forum 15 (1978), 343–350.Google Scholar