Difference between revisions of "Choquet boundary"
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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" />. | 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" />. | ||
− | 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" />. | + | 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" />. |
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. | 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. |
Revision as of 13:46, 6 March 2018
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) |
Choquet boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet_boundary&oldid=42920