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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]]].

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

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

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 [[#References|[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|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

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

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|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|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====
<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>