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The concept of the Bergman–Shilov boundary is credited to both S. Bergman and G.E. Shilov. Bergman, in 1931, introduced the concept of a distinguished boundary in connection with his studies of holomorphic functions of two complex variables [[#References|[a1]]]. He discovered that, for a large class of domains, a [[Holomorphic function|holomorphic function]] of two or more variables is completely determined by its values on a relatively small closed subset of the topological boundary of the domain. This phenomenon does not occur in function theory of one complex variable (cf. also [[Analytic function|Analytic function]]). On the other hand, in the 1940s Shilov introduced the concept of a minimal boundary, now called the Shilov boundary, in connection with his studies on commutative Banach algebras (cf. also [[Boundary (in the theory of uniform algebras)|Boundary (in the theory of uniform algebras)]]; [[Commutative Banach algebra|Commutative Banach algebra]]). Much of the content of this work can be found in [[#References|[a3]]].
 
The concept of the Bergman–Shilov boundary is credited to both S. Bergman and G.E. Shilov. Bergman, in 1931, introduced the concept of a distinguished boundary in connection with his studies of holomorphic functions of two complex variables [[#References|[a1]]]. He discovered that, for a large class of domains, a [[Holomorphic function|holomorphic function]] of two or more variables is completely determined by its values on a relatively small closed subset of the topological boundary of the domain. This phenomenon does not occur in function theory of one complex variable (cf. also [[Analytic function|Analytic function]]). On the other hand, in the 1940s Shilov introduced the concept of a minimal boundary, now called the Shilov boundary, in connection with his studies on commutative Banach algebras (cf. also [[Boundary (in the theory of uniform algebras)|Boundary (in the theory of uniform algebras)]]; [[Commutative Banach algebra|Commutative Banach algebra]]). Much of the content of this work can be found in [[#References|[a3]]].
  
In modern terminology, the Shilov boundary is defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103101.png" /> be a [[Compact space|compact space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103102.png" /> an algebra of continuous complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103103.png" /> which separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103104.png" /> (cf. [[Algebra of functions|Algebra of functions]]). A boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103105.png" /> is a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103106.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103107.png" /> such that
+
In modern terminology, the Shilov boundary is defined as follows: Let $  X $
 +
be a [[Compact space|compact space]] and $  {\mathcal F} $
 +
an algebra of continuous complex-valued functions on $  X $
 +
which separates the points of $  X $(
 +
cf. [[Algebra of functions|Algebra of functions]]). A boundary for $  {\mathcal F} $
 +
is a closed subset $  S $
 +
of $  X $
 +
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/b/b110/b110310/b1103108.png" /></td> </tr></table>
+
$$
 +
\left | {f ( x ) } \right | \leq  \max  \left \{ {\left | {f ( t ) } \right | } : {t \in S } \right \}
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b1103109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031010.png" />. The Shilov boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031012.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031013.png" />, is defined as the intersection of all boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031016.png" />. It was proved by Shilov [[#References|[a3]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031017.png" /> is non-empty and is a boundary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031018.png" />.
+
for all $  f \in {\mathcal F} $
 +
and $  x \in X $.  
 +
The Shilov boundary of $  X $
 +
for $  {\mathcal F} $,  
 +
denoted by $  S _  {\mathcal F}  ( X ) $,  
 +
is defined as the intersection of all boundaries $  S $
 +
of $  X $
 +
for $  {\mathcal F} $.  
 +
It was proved by Shilov [[#References|[a3]]] that $  S _  {\mathcal F}  ( X ) $
 +
is non-empty and is a boundary for $  {\mathcal F} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031019.png" /> is a bounded domain in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031020.png" />-dimensional [[Complex space|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031022.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031023.png" /> is the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031024.png" /> that are holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031025.png" /> (the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031026.png" />), then the Shilov boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031027.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031028.png" /> is called the Bergman–Shilov boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031029.png" />, usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031030.png" />. By the [[Maximum principle|maximum principle]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031031.png" /> is always a subset of the topological boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031032.png" />. However, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031033.png" />, for many domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031034.png" /> is a proper subset of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031035.png" />. For example, if
+
If $  D $
 +
is a bounded domain in an $  n $-
 +
dimensional [[Complex space|complex space]] $  \mathbf C  ^ {n} $,  
 +
$  n \geq  1 $,  
 +
and if $  {\mathcal F} = {\mathcal H} ( {\overline{D}\; } ) $
 +
is the set of all functions $  f $
 +
that are holomorphic on $  {\overline{D}\; } $(
 +
the closure of $  D $),  
 +
then the Shilov boundary of $  {\overline{D}\; } $
 +
for $  {\mathcal F} $
 +
is called the Bergman–Shilov boundary of $  D $,  
 +
usually denoted by $  B ( D ) $.  
 +
By the [[Maximum principle|maximum principle]], $  B ( D ) $
 +
is always a subset of the topological boundary of $  D $.  
 +
However, when $  n \geq  2 $,  
 +
for many domains $  B ( D ) $
 +
is a proper subset of the boundary of $  D $.  
 +
For example, if
  
<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/b/b110/b110310/b11031036.png" /></td> </tr></table>
+
$$
 +
D = \left \{ {( z,w ) \in \mathbf C  ^ {2} } : {\left | z \right | < 1,  \left | w \right | < 1 } \right \} ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031037.png" />, which is a two-dimensional closed subset of the three-dimensional boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031038.png" />. This is the case for a wide class of domains, including all analytic polyhedra (cf. [[Analytic polyhedron|Analytic polyhedron]]). On the other hand, if
+
then $  B ( D ) = \{ {( z,w ) } : {| z | = | w | = 1 } \} $,  
 +
which is a two-dimensional closed subset of the three-dimensional boundary of $  D $.  
 +
This is the case for a wide class of domains, including all analytic polyhedra (cf. [[Analytic polyhedron|Analytic polyhedron]]). On the other hand, if
  
<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/b/b110/b110310/b11031039.png" /></td> </tr></table>
+
$$
 +
D = \left \{ {( z,w ) \in \mathbf C  ^ {2} } : {\left | z \right |  ^ {2} + \left | w \right |  ^ {2} < 1 } \right \} ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031040.png" /> is equal to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031041.png" />. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031042.png" /> is strictly pseudo-convex, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031044.png" /> is a plurisubharmonic function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031045.png" /> (cf. also [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031046.png" /> is equal to the topological boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110310/b11031047.png" />. A fairly comprehensive treatment of the Shilov and Bergman–Shilov boundaries can be found in [[#References|[a2]]].
+
then $  B ( D ) $
 +
is equal to the boundary of $  D $.  
 +
More generally, if $  D $
 +
is strictly pseudo-convex, that is, $  D = \{ {z \in \mathbf C  ^ {n} } : {\phi ( z ) < 0 } \} $,  
 +
where $  \phi $
 +
is a plurisubharmonic function in a neighbourhood of $  {\overline{D}\; } $(
 +
cf. also [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]), then $  B ( D ) $
 +
is equal to the topological boundary of $  D $.  
 +
A fairly comprehensive treatment of the Shilov and Bergman–Shilov boundaries can be found in [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bergman,  "Über ausgezeichnete Randflächen in der Theorie der Functionen von Zwei komplexen Veränderlichen"  ''Math. Ann.'' , '''104'''  (1931)  pp. 611–636</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.A. Fuks,  "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Gel'fand,  D. Raikov,  G. Shilov,  "Commutative normed rings" , Chelsea  (1964)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bergman,  "Über ausgezeichnete Randflächen in der Theorie der Functionen von Zwei komplexen Veränderlichen"  ''Math. Ann.'' , '''104'''  (1931)  pp. 611–636</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.A. Fuks,  "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Gel'fand,  D. Raikov,  G. Shilov,  "Commutative normed rings" , Chelsea  (1964)  (In Russian)</TD></TR></table>

Revision as of 10:58, 29 May 2020


The concept of the Bergman–Shilov boundary is credited to both S. Bergman and G.E. Shilov. Bergman, in 1931, introduced the concept of a distinguished boundary in connection with his studies of holomorphic functions of two complex variables [a1]. He discovered that, for a large class of domains, a holomorphic function of two or more variables is completely determined by its values on a relatively small closed subset of the topological boundary of the domain. This phenomenon does not occur in function theory of one complex variable (cf. also Analytic function). On the other hand, in the 1940s Shilov introduced the concept of a minimal boundary, now called the Shilov boundary, in connection with his studies on commutative Banach algebras (cf. also Boundary (in the theory of uniform algebras); Commutative Banach algebra). Much of the content of this work can be found in [a3].

In modern terminology, the Shilov boundary is defined as follows: Let $ X $ be a compact space and $ {\mathcal F} $ an algebra of continuous complex-valued functions on $ X $ which separates the points of $ X $( cf. Algebra of functions). A boundary for $ {\mathcal F} $ is a closed subset $ S $ of $ X $ such that

$$ \left | {f ( x ) } \right | \leq \max \left \{ {\left | {f ( t ) } \right | } : {t \in S } \right \} $$

for all $ f \in {\mathcal F} $ and $ x \in X $. The Shilov boundary of $ X $ for $ {\mathcal F} $, denoted by $ S _ {\mathcal F} ( X ) $, is defined as the intersection of all boundaries $ S $ of $ X $ for $ {\mathcal F} $. It was proved by Shilov [a3] that $ S _ {\mathcal F} ( X ) $ is non-empty and is a boundary for $ {\mathcal F} $.

If $ D $ is a bounded domain in an $ n $- dimensional complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, and if $ {\mathcal F} = {\mathcal H} ( {\overline{D}\; } ) $ is the set of all functions $ f $ that are holomorphic on $ {\overline{D}\; } $( the closure of $ D $), then the Shilov boundary of $ {\overline{D}\; } $ for $ {\mathcal F} $ is called the Bergman–Shilov boundary of $ D $, usually denoted by $ B ( D ) $. By the maximum principle, $ B ( D ) $ is always a subset of the topological boundary of $ D $. However, when $ n \geq 2 $, for many domains $ B ( D ) $ is a proper subset of the boundary of $ D $. For example, if

$$ D = \left \{ {( z,w ) \in \mathbf C ^ {2} } : {\left | z \right | < 1, \left | w \right | < 1 } \right \} , $$

then $ B ( D ) = \{ {( z,w ) } : {| z | = | w | = 1 } \} $, which is a two-dimensional closed subset of the three-dimensional boundary of $ D $. This is the case for a wide class of domains, including all analytic polyhedra (cf. Analytic polyhedron). On the other hand, if

$$ D = \left \{ {( z,w ) \in \mathbf C ^ {2} } : {\left | z \right | ^ {2} + \left | w \right | ^ {2} < 1 } \right \} , $$

then $ B ( D ) $ is equal to the boundary of $ D $. More generally, if $ D $ is strictly pseudo-convex, that is, $ D = \{ {z \in \mathbf C ^ {n} } : {\phi ( z ) < 0 } \} $, where $ \phi $ is a plurisubharmonic function in a neighbourhood of $ {\overline{D}\; } $( cf. also Pseudo-convex and pseudo-concave), then $ B ( D ) $ is equal to the topological boundary of $ D $. A fairly comprehensive treatment of the Shilov and Bergman–Shilov boundaries can be found in [a2].

References

[a1] S. Bergman, "Über ausgezeichnete Randflächen in der Theorie der Functionen von Zwei komplexen Veränderlichen" Math. Ann. , 104 (1931) pp. 611–636
[a2] B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (In Russian)
[a3] I. Gel'fand, D. Raikov, G. Shilov, "Commutative normed rings" , Chelsea (1964) (In Russian)
How to Cite This Entry:
Bergman-Shilov boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bergman-Shilov_boundary&oldid=46014
This article was adapted from an original article by M. Stoll (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article