View source for Bergman-Shilov boundary

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

<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>

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" />.

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

<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>

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

<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>

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]]].

====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>