# 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)
How to Cite This Entry:
Choquet boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet_boundary&oldid=46340
This article was adapted from an original article by V.D. Pathak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article