Normal element

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a *-Algebra. An element 
  
    
      
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}}
  
 is called normal if it commutes with 
  
    
      
        
          a
          
            ∗
          
        
      
    
    {\displaystyle a^{*}}
  
, i.e. it satisfies the <a href="/facts/Equation/pxFzLAVR">equation</a> 
  
    
      
        a
        
          a
          
            ∗
          
        
        =
        
          a
          
            ∗
          
        
        a
      
    
    {\displaystyle aa^{*}=a^{*}a}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>The <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of normal elements is denoted by 
  
    
      
        
          
            
              A
            
          
          
            N
          
        
      
    
    {\displaystyle {\mathcal {A}}_{N}}
  
 or 
  
    
      
        N
        (
        
          
            A
          
        
        )
      
    
    {\displaystyle N({\mathcal {A}})}
  
.
</p><p>A special case of particular importance is the case where 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 is a <a href="/facts/Banach_algebra/3BTHdpg2">complete normed *-algebra</a>, that satisfies the C*-identity (
  
    
      
        
          ‖
          
            
              a
              
                ∗
              
            
            a
          
          ‖
        
        =
        
          
            ‖
            a
            ‖
          
          
            2
          
        
         
        ∀
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}
  
), which is called a <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>.
</p>
<h2 id="examples">Examples</h2>
<ul><li>Every <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint element</a> of a a *-algebra is normal.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li>Every <a href="/facts/Unitary_element/N6zYf8XN">unitary element</a> of a a *-algebra is normal.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li>
<li>If 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 is a C*-Algebra and 
  
    
      
        a
        ∈
        
          
            
              A
            
          
          
            N
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}_{N}}
  
 a normal element, then for every <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 on the <a href="/facts/Banach_algebra/3BTHdpg2">spectrum</a> of 
  
    
      
        a
      
    
    {\displaystyle a}
  
 the <a href="/facts/Continuous_functional_calculus/cJ2Nj9Bo">continuous functional calculus</a> defines another normal element 
  
    
      
        f
        (
        a
        )
      
    
    {\displaystyle f(a)}
  
.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul>
<h2 id="criteria">Criteria</h2>
<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a *-algebra. Then:
</p>
<ul><li>An element 
  
    
      
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}}
  
 is normal if and only if the *-<a href="/facts/Subalgebra/9PEHNF5r">subalgebra</a> generated by 
  
    
      
        a
      
    
    {\displaystyle a}
  
, meaning the smallest *-algebra containing 
  
    
      
        a
      
    
    {\displaystyle a}
  
, is commutative.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li>
<li>Every element 
  
    
      
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}}
  
 can be uniquely decomposed into a <a href="/facts/Real_and_imaginary_parts/2w7mqI7e">real and imaginary part</a>, which means there exist self-adjoint elements 
  
    
      
        
          a
          
            1
          
        
        ,
        
          a
          
            2
          
        
        ∈
        
          
            
              A
            
          
          
            s
            a
          
        
      
    
    {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}
  
, such that 
  
    
      
        a
        =
        
          a
          
            1
          
        
        +
        
          i
        
        
          a
          
            2
          
        
      
    
    {\displaystyle a=a_{1}+\mathrm {i} a_{2}}
  
, where 
  
    
      
        
          i
        
      
    
    {\displaystyle \mathrm {i} }
  
 denotes the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>. Exactly then 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is normal if 
  
    
      
        
          a
          
            1
          
        
        
          a
          
            2
          
        
        =
        
          a
          
            2
          
        
        
          a
          
            1
          
        
      
    
    {\displaystyle a_{1}a_{2}=a_{2}a_{1}}
  
, i.e. real and imaginary part commutate.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li></ul>
<h2 id="properties">Properties</h2>
<h3>In *-algebras</h3>
<p>Let 
  
    
      
        a
        ∈
        
          
            
              A
            
          
          
            N
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}_{N}}
  
 be a normal element of a *-algebra 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
. Then:
</p>
<ul><li>The adjoint element 
  
    
      
        
          a
          
            ∗
          
        
      
    
    {\displaystyle a^{*}}
  
 is also normal, since 
  
    
      
        a
        =
        (
        
          a
          
            ∗
          
        
        
          )
          
            ∗
          
        
      
    
    {\displaystyle a=(a^{*})^{*}}
  
 holds for the <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> *.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li></ul>
<h3>In C*-algebras</h3>
<p>Let 
  
    
      
        a
        ∈
        
          
            
              A
            
          
          
            N
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}_{N}}
  
 be a normal element of a C*-algebra 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
. Then:
</p>
<ul><li>It is 
  
    
      
        
          ‖
          
            a
            
              2
            
          
          ‖
        
        =
        
          
            ‖
            a
            ‖
          
          
            2
          
        
      
    
    {\displaystyle \left\|a^{2}\right\|=\left\|a\right\|^{2}}
  
, since for normal elements using the C*-identity 
  
    
      
        
          
            ‖
            
              a
              
                2
              
            
            ‖
          
          
            2
          
        
        =
        
          ‖
          
            (
            
              a
              
                2
              
            
            )
            (
            
              a
              
                2
              
            
            
              )
              
                ∗
              
            
          
          ‖
        
        =
        
          ‖
          
            (
            
              a
              
                ∗
              
            
            a
            
              )
              
                ∗
              
            
            (
            
              a
              
                ∗
              
            
            a
            )
          
          ‖
        
        =
        
          
            ‖
            
              
                a
                
                  ∗
                
              
              a
            
            ‖
          
          
            2
          
        
        =
        
          
            (
            
              
                ‖
                a
                ‖
              
              
                2
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle \left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}}
  
 holds.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li>
<li>Every normal element is a normaloid element, i.e. the <a href="/facts/Spectral_radius/j8T2xrLu">spectral radius</a> 
  
    
      
        r
        (
        a
        )
      
    
    {\displaystyle r(a)}
  
 equals the norm of 
  
    
      
        a
      
    
    {\displaystyle a}
  
, i.e. 
  
    
      
        r
        (
        a
        )
        =
        
          ‖
          a
          ‖
        
      
    
    {\displaystyle r(a)=\left\|a\right\|}
  
.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> This follows from the <a href="/facts/Spectral_radius/j8T2xrLu">spectral radius formula</a> by repeated application of the previous property.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li>
<li>A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of 
  
    
      
        a
      
    
    {\displaystyle a}
  
 to 
  
    
      
        a
      
    
    {\displaystyle a}
  
.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Normal_matrix/WVKxrEK9">Normal matrix</a></li>
<li><a href="/facts/Normal_operator/lwtz7tuF">Normal operator</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Dixmier, Jacques (1977). <i>C*-algebras</i>. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7204-0762-1. English translation of <i>Les C*-algèbres et leurs représentations</i> (in French). Gauthier-Villars. 1969.</li>
<li>Heuser, Harro (1982). <i>Functional analysis</i>. Translated by Horvath, John. John Wiley & Sons Ltd. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-10069-2.</li>
<li>Werner, Dirk (2018). <i>Funktionalanalysis</i> (in German) (8 ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-662-55407-4.</li></ul>

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

<ol>
<li id="fn:1"><p>Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Dixmier 1977, p. 5. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Dixmier 1977, p. 5. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Dixmier 1977, pp. 3–4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Werner 2018, p. 518. - Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Heuser 1982, p. 390. - Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Werner 2018, pp. 284–285, 518. - Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
</ol>

Normal element open-in-new

Normal element