J Austral Math Soc Ser A 45 pp1--10, 1988.

Analytic Functions Which Operate on Homogeneous Algebras

J. A. Ward

Abstract

It is well known that a complex-valued function f, analytic on some open set W, extends to any commutative Banach algebra B so that the action of f on B commutes with the action of the Gelfand transformation. In this paper, it is shown that if B is a homogenous convolution Banach algebra over any compact group and if 0 Î W is a fixed point of f, 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 f to B and to certain operator algebras.

1980 AMS Subject Classification: primary 43A10; secondary 43A15, 46H30

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Authors

J. A. Ward

School of Mathematics and Physical Sciences, Murdoch University, Perth, Western Australia 6153, Australia.