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Choquet Boundary for Real Function Algebras

Published online by Cambridge University Press:  20 November 2018

S. H. Kulkarni  and
S. Arundhathi
S. H. Kulkarni
Affiliation:
Indian Institute of Technology, Madras, India
S. Arundhathi
Affiliation:
Indian Institute of Technology, Madras, India
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The concepts of Choquet boundary and Shilov boundary are well-established in the context of a complex function algebra (see [2] for example). There have been a few attempts to develop the concept of a Shilov boundary for real algebras, [4], [6] and [7]. But there seems to be none to develop the concept of Choquet boundary for real algebras.

The aim of this paper is to develop the theory of Choquet boundary of a real function algebra (see Definition (1.8)) along the lines of the corresponding theory for a complex function algebra.

In the first section we define a real-part representing measure for a continuous linear functional ϕ on a real function algebra A with the property ║ϕ║ = 1 = ϕ(1). The elements of A are functions on a compact, Hausdorff space X. The Choquet boundary is then defined as the set of those points x ∊ X such that the real part of the evaluation functional, Re(ex), has a unique real part representing measure.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 5 , 01 October 1988 , pp. 1084 - 1104
Copyright
Copyright © Canadian Mathematical Society 1988

References

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