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Analytic functions which operate on homogeneous algebras

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

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
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Abstract

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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

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