Unitary element

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a *-algebra with <a href="/facts/Identity_element/9nss3xHY">unit</a> 
  
    
      
        e
      
    
    {\displaystyle e}
  
. An element 
  
    
      
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}}
  
 is called unitary if 
  
    
      
        a
        
          a
          
            ∗
          
        
        =
        
          a
          
            ∗
          
        
        a
        =
        e
      
    
    {\displaystyle aa^{*}=a^{*}a=e}
  
. In other words, if 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is invertible and 
  
    
      
        
          a
          
            −
            1
          
        
        =
        
          a
          
            ∗
          
        
      
    
    {\displaystyle a^{-1}=a^{*}}
  
 holds, then 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is unitary.<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 unitary elements is denoted by 
  
    
      
        
          
            
              A
            
          
          
            U
          
        
      
    
    {\displaystyle {\mathcal {A}}_{U}}
  
 or 
  
    
      
        U
        (
        
          
            A
          
        
        )
      
    
    {\displaystyle U({\mathcal {A}})}
  
.
</p><p>A special case from particular importance is the case where 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 is a <a href="/facts/Banach_algebra/3BTHdpg2">complete normed *-algebra</a>. This algebra satisfies the C*-identity (
  
    
      
        
          ‖
          
            
              a
              
                ∗
              
            
            a
          
          ‖
        
        =
        
          
            ‖
            a
            ‖
          
          
            2
          
        
         
        ∀
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}
  
) and is called a <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>.
</p>
<h2 id="criteria">Criteria</h2>
<ul><li>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a <a href="/facts/Algebra_over_a_field/8T8ulWJb">unital</a> C*-algebra and 
  
    
      
        a
        ∈
        
          
            
              A
            
          
          
            N
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}_{N}}
  
 a <a href="/facts/Normal_element/RHIIdbaQ">normal</a> element. Then, 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is unitary if the <a href="/facts/Banach_algebra/3BTHdpg2">spectrum</a> 
  
    
      
        σ
        (
        a
        )
      
    
    {\displaystyle \sigma (a)}
  
 consists only of elements of the <a href="/facts/Circle_group/kuzdWtHo">circle group</a> 
  
    
      
        
          T
        
      
    
    {\displaystyle \mathbb {T} }
  
, i.e. 
  
    
      
        σ
        (
        a
        )
        ⊆
        
          T
        
        =
        {
        λ
        ∈
        
          C
        
        ∣
        
          |
        
        λ
        
          |
        
        =
        1
        }
      
    
    {\displaystyle \sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li></ul>
<h2 id="examples">Examples</h2>
<ul><li>The unit 
  
    
      
        e
      
    
    {\displaystyle e}
  
 is unitary.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul>
<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a unital C*-algebra, then:
</p>
<ul><li>Every <a href="/facts/Projection_(mathematics)/QxxC9XTp">projection</a>, i.e. every element 
  
    
      
        a
        ∈
        
          
            A
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}}
  
 with 
  
    
      
        a
        =
        
          a
          
            ∗
          
        
        =
        
          a
          
            2
          
        
      
    
    {\displaystyle a=a^{*}=a^{2}}
  
, is unitary. For the spectrum of a projection consists of at most 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 and 
  
    
      
        1
      
    
    {\displaystyle 1}
  
, as follows from the <a href="/facts/Continuous_functional_calculus/cJ2Nj9Bo">continuous functional calculus</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li>
<li>If 
  
    
      
        a
        ∈
        
          
            
              A
            
          
          
            N
          
        
      
    
    {\displaystyle a\in {\mathcal {A}}_{N}}
  
 is a normal element of a C*-algebra 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
, then for every <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 on the spectrum 
  
    
      
        σ
        (
        a
        )
      
    
    {\displaystyle \sigma (a)}
  
 the continuous functional calculus defines an unitary element 
  
    
      
        f
        (
        a
        )
      
    
    {\displaystyle f(a)}
  
, if 
  
    
      
        f
        (
        σ
        (
        a
        )
        )
        ⊆
        
          T
        
      
    
    {\displaystyle f(\sigma (a))\subseteq \mathbb {T} }
  
.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ul>
<h2 id="properties">Properties</h2>
<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a unital *-algebra and 
  
    
      
        a
        ,
        b
        ∈
        
          
            
              A
            
          
          
            U
          
        
      
    
    {\displaystyle a,b\in {\mathcal {A}}_{U}}
  
. Then:
</p>
<ul><li>The element 
  
    
      
        a
        b
      
    
    {\displaystyle ab}
  
 is unitary, since 
  
    
      
        (
        (
        a
        b
        
          )
          
            ∗
          
        
        
          )
          
            −
            1
          
        
        =
        (
        
          b
          
            ∗
          
        
        
          a
          
            ∗
          
        
        
          )
          
            −
            1
          
        
        =
        (
        
          a
          
            ∗
          
        
        
          )
          
            −
            1
          
        
        (
        
          b
          
            ∗
          
        
        
          )
          
            −
            1
          
        
        =
        a
        b
      
    
    {\textstyle ((ab)^{*})^{-1}=(b^{*}a^{*})^{-1}=(a^{*})^{-1}(b^{*})^{-1}=ab}
  
. In particular, 
  
    
      
        
          
            
              A
            
          
          
            U
          
        
      
    
    {\displaystyle {\mathcal {A}}_{U}}
  
 forms a <a href="/facts/Multiplicative_group/G209dICf">multiplicative group</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li>
<li>The element 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is normal.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li>
<li>The adjoint element 
  
    
      
        
          a
          
            ∗
          
        
      
    
    {\displaystyle a^{*}}
  
 is also unitary, since 
  
    
      
        a
        =
        (
        
          a
          
            ∗
          
        
        
          )
          
            ∗
          
        
      
    
    {\displaystyle a=(a^{*})^{*}}
  
 holds for the <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> *.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li>
<li>If 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 is a C*-algebra, 
  
    
      
        a
      
    
    {\displaystyle a}
  
 has norm 1, i.e. 
  
    
      
        
          ‖
          a
          ‖
        
        =
        1
      
    
    {\displaystyle \left\|a\right\|=1}
  
.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Unitary_matrix/8fsxv3KD">Unitary matrix</a></li>
<li><a href="/facts/Unitary_operator/rkypfQQ9">Unitary operator</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Blackadar, Bruce (2006). <i>Operator Algebras. Theory of C*-Algebras and von Neumann Algebras</i>. Berlin/Heidelberg: Springer. pp. 57, 63. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-28486-9.</li>
<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>Kadison, Richard V.; Ringrose, John R. (1983). <i><a href="/facts/Fundamentals_of_the_Theory_of_Operator_Algebras/8N9qf6sl">Fundamentals of the Theory of Operator Algebras</a>. Volume 1 Elementary Theory</i>. New York/London: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-393301-3.</li></ul>

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

<ol>
<li id="fn:1"><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:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><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:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Kadison & Ringrose 1983, p. 271. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Dixmier 1977, pp. 4–5. - 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>Blackadar 2006, pp. 57, 63. - Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. pp. 57, 63. ISBN 3-540-28486-9. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Kadison & Ringrose 1983, p. 271. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><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:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Dixmier 1977, pp. 4–5. - 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>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:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Dixmier 1977, p. 9. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
</ol>

Unitary element open-in-new

Unitary element