Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T04:16:14.131Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic functions which operate on homogeneous algebras

Part of: Abstract harmonic analysis Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  09 April 2009

J. A. Ward
J. A. Ward
Affiliation:
School of Mathematical and Physical SciencesMurdoch UniversityPerth, Western Australia 6153, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that a complex-valued function ø, analytic on some open set Ω, extends to any commutative Banach algebra B so that the action of ø on B commutes with the action of the Gelfand transformation. In this paper, it is shown that if B is a homogeneous convolution Banach algebra over any compact group and if 0 ∈ Ω is a fixed point of ø, then a similar result holds, with the Gelfand transformation replaced by the Fourier-Stieltjes transformation. Care is required, in that discussion of this relation usually requires simultaneous consideration of the extension of ø to B and to certain operator algebras.

MSC classification

Secondary: 43A10: Measure algebras on groups, semigroups, etc. 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 46H30: Functional calculus in topological algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 1 - 10
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Ahlfors, L. V., Complex analysis, 2nd ed., (McGraw-Hill, New York, 1966).Google Scholar
[2]Edwards, R. E., Fourier series, a modern introduction, (Holt, Rinehart and Winston, New York, 1967).Google Scholar
[3]Fountain, J. B., Ramsay, R. W. and Williamson, J. H., ‘Functions of measures on compact groups’, Proc. Roy. Irish Acad. 77 (1977), 235–251.Google Scholar
[4]Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Vol. I and II, (Springer-Verlag, Berlin, 1966, 1970).Google Scholar
[5]Hille, E. and Phillips, R. S., Functional analysis and semi-groups, (Amer. Math. Soc. Colloquium Publication, Vol. 31, 1957).Google Scholar
[6]Larsen, R., Banach algebras, an introduction, (Marcel Dekker, New York, 1973).Google Scholar
[7]Loomis, L., An introduction to abstract harmonic analysis, (Van Nostrand, New York, 1953).Google Scholar
[8]Reiter, H. J., L1-algebras and Segal algebras, (Lecture Notes in Mathematics, vol. 231, Springer-Verlag, Berlin and New York, 1971).CrossRefGoogle Scholar
[9]Rickart, C. E., General theory of Banach algebras, (Van Nostrand, New York, 1960).Google Scholar
[10]Segal, I. E., ‘The group algebra of a locally compact group’, Trans. Amer. Math. Soc. 61 (1947), 69–105.Google Scholar
[11]Wang, H. C., Homogeneous Banach algebras, (Lecture Notes in Pure and Appl. Mathematics, Dekker, 1977).Google Scholar
[12]Ward, J. A., ‘Ideal structure of operator and measure algebras,’ Monatsh. Math. 95 (1983), 159–172.CrossRefGoogle Scholar
[13]Ward, J. A., ‘Characterisation of homogeneous spaces and their norms’, Pacific J. Math. 114 (1984), 481–495.CrossRefGoogle Scholar
[14]Ward, J. A., ‘Closed ideals of homogeneous algebras’, Monatsh. Math. 96 (1983), 317–324.CrossRefGoogle Scholar
[15]Wiener, N., The Fourier integral and certain of its applications, (Cambridge University Press, New York, 1933).Google Scholar