Choquet boundary

Let $ X $ be a compact Hausdorff space (cf. also Compact space), let $ C ( X ) $ be the Banach algebra of all complex-valued continuous functions on $ X $ with the supremum norm and let $ A $ be a linear subspace of $ C ( X ) $ containing the constant functions on $ X $. For $ t \in X $, let $ \tau _ {t} \in A ^ {*} $ be defined by $ \tau _ {t} ( f ) = f ( t ) $ for all $ f \in A $ and let $ K ( A ) $ denote the set $ \{ {x ^ {*} \in A ^ {*} } : {\| {x ^ {*} } \| = x ^ {*} ( 1 ) = 1 } \} $.

The Choquet boundary for $ A $ is defined as the set

$$ { \mathop{\rm Ch} } ( A ) = \left \{ {t \in X } : {\tau _ {t} \in { \mathop{\rm ext} } ( K ( A ) ) } \right \} , $$

where $ { \mathop{\rm ext} } ( K ( A ) ) $ denotes the set of extreme points of $ K ( A ) $.

Other relevant concepts involved in the study of the Choquet boundary are the boundary and the Shilov boundary. A boundary for $ A $ is a subset $ E $ of $ X $ such that for each $ f \in A $, there exists a $ t \in E $ such that $ | {f ( t ) } | = \| f \| _ \infty $( cf. also Boundary (in the theory of uniform algebras)). If there is a smallest closed boundary for $ A $, then it is called the Shilov boundary for $ A $; it is denoted by $ \delta A $.

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

It is clear from the above discussion that $ { \mathop{\rm Ch} } ( A ( D ) ) = \delta ( A ( D ) ) = \Gamma $. Also, $ { \mathop{\rm Ch} } ( C ( X ) ) = X $ and $ { \mathop{\rm Ch} } ( P ( D _ {2} ) ) = \delta ( P ( D _ {2} ) ) = \Gamma _ {2} $, where $ D _ {2} = D \times D $, $ \Gamma _ {2} = \Gamma \times \Gamma $, and $ P ( D _ {2} ) $ is the uniform closure on $ D _ {2} $ of the algebra of all polynomials in two complex variables (cf. also Uniform space).

In general, $ { \mathop{\rm Ch} } ( A ) $ is a boundary of $ A $ and hence $ { \mathop{\rm Ch} } ( A ) \neq \emptyset $. If, in addition, $ A $ separates the points of $ X $( cf. Uniform algebra), then $ { \mathop{\rm Ch} } ( A ) $ is dense in $ \delta A $. Also, in this case, $ t \in { \mathop{\rm Ch} } ( A ) $ if and only if the $ \varepsilon _ {t} $( the unit mass concentrated at $ t $) is the unique representing measure for $ \tau _ {t} $. This equivalent description is used as a definition of Choquet boundary in [a1].

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

$$ \rho A = \left \{ {t \in X } : {\exists f \in A: \left | {f ( t ) } \right | < \left | {f ( s ) } \right | , \forall s \in X \setminus \{ t \} } \right \} . $$

For any such algebra $ A $, $ \rho A \subseteq { \mathop{\rm Ch} } ( A ) \subseteq \delta A $ and if, in addition, $ X $ is metrizable (cf. Metrizable space), then $ { \mathop{\rm Ch} } ( A ) $ is a $ G _ \delta $- set. However, if $ X $ is not metrizable, then the following example [a5] shows that $ { \mathop{\rm Ch} } ( A ) $ need not even be a Borel set. Let $ \Lambda $ be an uncountable index set and for each $ \lambda \in \Lambda $, let $ A _ \lambda = \{ {f \in A ( D ) } : {f ( 0 ) = f ( 1 ) } \} $ and $ B _ \lambda = \Gamma - \{ 1 \} $. Then for $ U = \otimes \{ {A _ \lambda } : {\lambda \in \Lambda } \} $ one has $ { \mathop{\rm Ch} } ( U ) = \prod \{ {B _ \lambda } : {\lambda \in \Lambda } \} $, which is not a Borel set since $ \Lambda $ is uncountable.

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

i) $ t \in { \mathop{\rm Ch} } ( A ) $;

ii) for each open neighbourhood $ U $ of $ t $ there is an $ f \in A $ such that $ \| f \| _ \infty = 1 $ and $ | {f ( s ) } | < 1 $ for all $ s \in X \setminus U $;

iii) there exists a family of peak sets $ \{ E _ {f _ \alpha } \} $ for $ A $ such that

$$ \cap _ \alpha E _ {f _ \alpha } = \{ t \} , $$

where, for $ f _ \alpha \in A $,

$$ E _ {f _ \alpha } = \left \{ {t \in X } : {\left | {f _ \alpha ( t ) } \right | = \left \| {f _ \alpha } \right \| _ \infty } \right \} ; $$

iv) given $ 0 < \alpha < \beta < 1 $, if $ U $ is an open neighbourhood of $ t $, then there is an $ f \in A $ such that $ \| f \| _ \infty < 1 $, $ | {f ( t ) } | > \beta $ and $ | {f ( s ) } | < \alpha $ for $ s \in X \setminus U $.

If $ A $ is a uniform algebra and $ X $ is metrizable, then $ { \mathop{\rm Ch} } ( A ) = \rho A $.

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

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 $ A $ is a subalgebra of $ C ( X ) $ separating points and containing the constants and if $ T : A \rightarrow A $ is a linear isometric mapping (linear isometry) of $ A $ onto $ A $ such that $ T ( 1 ) = 1 $, then one can show that $ T $ is an algebra isometry of $ A $ onto $ A $, [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)