Shilov boundary

<h2 id="precise-definition-and-existence">Precise definition and existence</h2>
<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a <a href="/facts/Commutative/WaYxJLxd">commutative</a> <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a> and let 
  
    
      
        Δ
        
          
            A
          
        
      
    
    {\displaystyle \Delta {\mathcal {A}}}
  
 be its <a href="/facts/Structure_space/3BTHdpg2">structure space</a> equipped with the <a href="/facts/Relative_topology/NjU6z4Ca">relative</a> <a href="/facts/Weak_topology/XJASE9Up">weak*-topology</a> of the <a href="/facts/Continuous_dual_space/4Uw4Knz1">dual</a> 
  
    
      
        
          
            
              A
            
          
          
            ∗
          
        
      
    
    {\displaystyle {\mathcal {A}}^{*}}
  
. A closed (in this topology) subset 
  
    
      
        F
      
    
    {\displaystyle F}
  
 of 
  
    
      
        Δ
        
          
            A
          
        
      
    
    {\displaystyle \Delta {\mathcal {A}}}
  
 is called a boundary of 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 if 
  
    
      
        
          max
          
            f
            ∈
            Δ
            
              
                A
              
            
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        =
        
          max
          
            f
            ∈
            F
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
      
    
    {\textstyle \max _{f\in \Delta {\mathcal {A}}}|f(x)|=\max _{f\in F}|f(x)|}
  
 for all 
  
    
      
        x
        ∈
        
          
            A
          
        
      
    
    {\displaystyle x\in {\mathcal {A}}}
  
.
The set 
  
    
      
        S
        =
        ⋂
        {
        F
        :
        F
        
           is a boundary of 
        
        
          
            A
          
        
        }
      
    
    {\textstyle S=\bigcap \{F:F{\text{ is a boundary of }}{\mathcal {A}}\}}
  
 is called the Shilov boundary. It has been proved by Shilov<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> that 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a boundary of 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
.
</p><p>Thus one may also say that Shilov boundary is the unique set 
  
    
      
        S
        ⊂
        Δ
        
          
            A
          
        
      
    
    {\displaystyle S\subset \Delta {\mathcal {A}}}
  
 which satisfies
</p>
<ol><li>
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a boundary of 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
, and</li>
<li>whenever 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is a boundary of 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
, then 
  
    
      
        S
        ⊂
        F
      
    
    {\displaystyle S\subset F}
  
.</li></ol>
<h2 id="examples">Examples</h2>
<p>Let 
  
    
      
        
          D
        
        =
        {
        z
        ∈
        
          C
        
        :
        
          |
        
        z
        
          |
        
        <
        1
        }
      
    
    {\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}
  
 be the <a href="/facts/Open_unit_disc/phrUu7yC">open unit disc</a> in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> and let 
  
    
      
        
          
            A
          
        
        =
        
          H
          
            ∞
          
        
        (
        
          D
        
        )
        ∩
        
          
            C
          
        
        (
        
          
            
              
                D
              
              ¯
            
          
        
        )
      
    
    {\displaystyle {\mathcal {A}}=H^{\infty }(\mathbb {D} )\cap {\mathcal {C}}({\bar {\mathbb {D} }})}
  
 be the <a href="/facts/Disc_algebra/9QVWbISL">disc algebra</a>, i.e. the functions <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> in 
  
    
      
        
          D
        
      
    
    {\displaystyle \mathbb {D} }
  
 and <a href="/facts/Continuous_function/sKbl02pB">continuous</a> in the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of 
  
    
      
        
          D
        
      
    
    {\displaystyle \mathbb {D} }
  
 with <a href="/facts/Supremum_norm/5yIkjNE3">supremum norm</a> and usual algebraic operations. Then 
  
    
      
        Δ
        
          
            A
          
        
        =
        
          
            
              
                D
              
              ¯
            
          
        
      
    
    {\displaystyle \Delta {\mathcal {A}}={\bar {\mathbb {D} }}}
  
 and 
  
    
      
        S
        =
        {
        
          |
        
        z
        
          |
        
        =
        1
        }
      
    
    {\displaystyle S=\{|z|=1\}}
  
.
</p>

<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Bergman-Shilov_boundary">"Bergman-Shilov boundary"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>
<h2 id="notes">Notes</h2>

<h2 id="see-also">See also</h2>
<ul><li>James boundary</li>
<li><a href="/facts/Furstenberg_boundary/VTW6fFtn">Furstenberg boundary</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: Functional analysis and semigroups. -- AMS, Providence 1957. <a href="/wiki/Einar_Hille" target="_blank">/wiki/Einar_Hille</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Shilov boundary open-in-new

Shilov boundary