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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102001.png" /> be a compact [[Hausdorff space|Hausdorff space]] (cf. also [[Compact space|Compact space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102002.png" /> be the [[Banach algebra|Banach algebra]] of all complex-valued continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102003.png" /> with the supremum norm and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102004.png" /> be a linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102005.png" /> containing the constant functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102006.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102007.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102008.png" /> be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c1102009.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020010.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020011.png" /> denote the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020012.png" />.
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$#C+1 = 119 : ~/encyclopedia/old_files/data/C110/C.1100200 Choquet boundary
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The Choquet boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020013.png" /> is defined as the set
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020014.png" /></td> </tr></table>
+
Let  $  X $
 +
be a compact [[Hausdorff space|Hausdorff space]] (cf. also [[Compact space|Compact space]]), let  $  C ( X ) $
 +
be the [[Banach algebra|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 } \} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020015.png" /> denotes the set of extreme points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020016.png" />.
+
The Choquet boundary for  $  A $
 +
is defined as the set
  
Other relevant concepts involved in the study of the Choquet boundary are the boundary and the [[Shilov boundary]]. A boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020017.png" /> is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020019.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020020.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020022.png" /> (cf. also [[Boundary (in the theory of uniform algebras)|Boundary (in the theory of uniform algebras)]]). If there is a smallest closed boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020023.png" />, then it is called the Shilov boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020024.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020025.png" />.
+
$$
 +
{ \mathop{\rm Ch} } ( A ) = \left \{ {t \in X } : {\tau _ {t} \in { \mathop{\rm ext} } ( K ( A ) ) } \right \} ,
 +
$$
  
The motivation for these concepts comes from the classical theory of analytic functions (cf. also [[Analytic function|Analytic function]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020026.png" /> denotes the closed unit disc and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020027.png" /> is the linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020028.png" /> consisting of all complex-valued functions that are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020029.png" /> and analytic inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020030.png" />, then, by the [[Maximum-modulus principle|maximum-modulus principle]], for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020031.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020032.png" /> (the unit circle) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020033.png" />. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020034.png" /> is the smallest closed set having this property. A natural question to ask is: Given an arbitrary linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020036.png" />, does there exist a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020037.png" /> having properties similar to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020038.png" />? Investigations in this direction have led to the introduction of the above concepts.
+
where  $  { \mathop{\rm ext} } ( K ( A ) ) $
 +
denotes the set of extreme points of $  K ( A ) $.
  
It is clear from the above discussion that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020039.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020044.png" /> is the uniform closure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020045.png" /> of the algebra of all polynomials in two complex variables (cf. also [[Uniform space|Uniform space]]).
+
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)|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 $.
  
In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020046.png" /> is a boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020047.png" /> and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020048.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020049.png" /> separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020050.png" /> (cf. [[Uniform algebra|Uniform algebra]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020051.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020052.png" />. Also, in this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020053.png" /> if and only if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020054.png" /> (the unit mass concentrated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020055.png" />) is the unique representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020056.png" />. This equivalent description is used as a definition of Choquet boundary in [[#References|[a1]]].
+
The motivation for these concepts comes from the classical theory of analytic functions (cf. also [[Analytic function|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|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.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020057.png" /> is a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020058.png" /> containing the constants and separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020059.png" />, then the Bishop boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020060.png" /> can be defined as the set of all peak points for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020061.png" />, that is, the set
+
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|Uniform space]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020062.png" /></td> </tr></table>
+
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|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 [[#References|[a1]]].
  
For any such algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020064.png" /> and if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020065.png" /> is metrizable (cf. [[Metrizable space|Metrizable space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020066.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020068.png" />-set. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020069.png" /> is not metrizable, then the following example [[#References|[a5]]] shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020070.png" /> need not even be a [[Borel set|Borel set]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020071.png" /> be an uncountable index set and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020072.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020074.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020075.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020076.png" />, which is not a Borel set since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020077.png" /> is uncountable.
+
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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020078.png" /> is a uniform algebra (i.e. a closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020079.png" /> containing the constants and separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020080.png" />), then the following are equivalent
+
$$
 +
\rho A = \left \{ {t \in X } : {\exists f \in A: \left | {f ( t ) } \right | < \left | {f ( s ) } \right | , \forall s \in X \setminus  \{ t \} } \right \} .
 +
$$
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020081.png" />;
+
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|Metrizable space]]), then  $  { \mathop{\rm Ch} } ( A ) $
 +
is a  $  G _  \delta  $-
 +
set. However, if  $  X $
 +
is not metrizable, then the following example [[#References|[a5]]] shows that  $  { \mathop{\rm Ch} } ( A ) $
 +
need not even be a [[Borel set|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.
  
ii) for each open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020083.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020084.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020086.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020087.png" />;
+
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
  
iii) there exists a family of peak sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020088.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020089.png" /> such that
+
i) $  t \in { \mathop{\rm Ch} } ( A ) $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020090.png" /></td> </tr></table>
+
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 $;
  
where, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020091.png" />,
+
iii) there exists a family of peak sets  $  \{ E _ {f _  \alpha  } \} $
 +
for $  A $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020092.png" /></td> </tr></table>
+
$$
 +
\cap _  \alpha  E _ {f _  \alpha  } = \{ t \} ,
 +
$$
  
iv) given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020093.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020094.png" /> is an open neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020095.png" />, then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020096.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c11020099.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200100.png" />.
+
where, for  $  f _  \alpha  \in A $,
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200101.png" /> is a uniform algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200102.png" /> is metrizable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200103.png" />.
+
$$
 +
E _ {f _  \alpha  } = \left \{ {t \in X } : {\left | {f _  \alpha  ( t ) } \right | = \left \| {f _  \alpha  } \right \| _  \infty  } \right \} ;
 +
$$
  
The concept of Choquet boundary can be extended to any arbitrary commutative Banach algebra via Gel'fand theory. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200104.png" /> is any commutative Banach algebra (cf. [[Commutative Banach algebra|Commutative Banach algebra]]) with identity, then its [[Maximal ideal|maximal ideal]] space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200105.png" /> is compact Hausdorff and the [[Gel'fand representation|Gel'fand representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200106.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200107.png" /> is a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200108.png" /> separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200109.png" /> and containing the constants. Hence, one can define the Choquet boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200110.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200111.png" />.
+
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|Commutative Banach algebra]]) with identity, then its [[Maximal ideal|maximal ideal]] space $  riangle ( A ) $
 +
is compact Hausdorff and the [[Gel'fand representation|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 [[#References|[a2]]].
 
The concept of Choquet boundary has been extended to real function algebras in [[#References|[a2]]].
  
The notion of Choquet boundary is useful in characterizing onto linear isometries of certain function spaces. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200112.png" /> is a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200113.png" /> separating points and containing the constants and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200114.png" /> is a linear [[Isometric mapping|isometric mapping]] (linear isometry) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200115.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200116.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200117.png" />, then one can show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200118.png" /> is an algebra isometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200119.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110200/c110200120.png" />, [[#References|[a3]]], p. 243.
+
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|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 $,  
 +
[[#References|[a3]]], p. 243.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Browder,  "Introduction to function algebras" , W.A. Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.H. Kulkarni,  B.V. Limaye,  "Real function algebras" , M. Dekker  (1992)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Larsen,  "Banach algebras: an introduction" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.L. Stout,  "The theory of uniform algebras" , Bogden and Quigley  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Browder,  "Introduction to function algebras" , W.A. Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.H. Kulkarni,  B.V. Limaye,  "Real function algebras" , M. Dekker  (1992)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Larsen,  "Banach algebras: an introduction" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.L. Stout,  "The theory of uniform algebras" , Bogden and Quigley  (1971)</TD></TR></table>

Revision as of 16:44, 4 June 2020


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=42920
This article was adapted from an original article by V.D. Pathak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article