Choquet boundary

Let be a compact Hausdorff space (cf. also Compact space), let be the Banach algebra of all complex-valued continuous functions on with the supremum norm and let be a linear subspace of containing the constant functions on . For , let be defined by for all and let denote the set .

The Choquet boundary for is defined as the set

where denotes the set of extreme points of .

Other relevant concepts involved in the study of the Choquet boundary are the boundary and the Shilov boundary. A boundary for is a subset of such that for each , there exists a such that (cf. also Boundary (in the theory of uniform algebras)). If there is a smallest closed boundary for , then it is called the Shilov boundary for ; it is denoted by .

The motivation for these concepts comes from the classical theory of analytic functions (cf. also Analytic function). If denotes the closed unit disc and is the linear subspace of consisting of all complex-valued functions that are continuous on and analytic inside , then, by the maximum-modulus principle, for each there exists a (the unit circle) such that . In fact, is the smallest closed set having this property. A natural question to ask is: Given an arbitrary linear subspace of , does there exist a subset of having properties similar to ? Investigations in this direction have led to the introduction of the above concepts.

It is clear from the above discussion that . Also, and , where , , and is the uniform closure on of the algebra of all polynomials in two complex variables (cf. also Uniform space).

In general, is a boundary of and hence . If, in addition, separates the points of (cf. Uniform algebra), then is dense in . Also, in this case, if and only if the (the unit mass concentrated at ) is the unique representing measure for . This equivalent description is used as a definition of Choquet boundary in [a1].

If is a subalgebra of containing the constants and separating the points of , then the Bishop boundary for can be defined as the set of all peak points for , that is, the set

For any such algebra , and if, in addition, is metrizable (cf. Metrizable space), then is a -set. However, if is not metrizable, then the following example [a5] shows that need not even be a Borel set. Let be an uncountable index set and for each , let and . Then for one has , which is not a Borel set since is uncountable.

If is a uniform algebra (i.e. a closed subalgebra of containing the constants and separating the points of ), then the following are equivalent

i) ;

ii) for each open neighbourhood of there is an such that and for all ;

iii) there exists a family of peak sets for such that

where, for ,

iv) given , if is an open neighbourhood of , then there is an such that , and for .

If is a uniform algebra and is metrizable, then .

The concept of Choquet boundary can be extended to any arbitrary commutative Banach algebra via Gel'fand theory. If is any commutative Banach algebra (cf. Commutative Banach algebra) with identity, then its maximal ideal space is compact Hausdorff and the Gel'fand representation of is a subalgebra of separating the points of and containing the constants. Hence, one can define the Choquet boundary of as .

The concept of Choquet boundary has been extended to real function algebras in [a2].

The notion of Choquet boundary is useful in characterizing onto linear isometries of certain function spaces. In particular, if is a subalgebra of separating points and containing the constants and if is a linear isometric mapping (linear isometry) of onto such that , then one can show that is an algebra isometry of onto , [a3], p. 243.

References

[a1]	A. Browder, "Introduction to function algebras" , W.A. Benjamin (1969)
[a2]	S.H. Kulkarni, B.V. Limaye, "Real function algebras" , M. Dekker (1992)
[a3]	R. Larsen, "Banach algebras: an introduction" , M. Dekker (1973)
[a4]	R.R. Phelps, "Lectures on Choquet's theorem" , v. Nostrand (1966)
[a5]	E.L. Stout, "The theory of uniform algebras" , Bogden and Quigley (1971)