*-algebra

<h2 id="definitions">Definitions</h2>
<h3>*-ring</h3>

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a *-ring is a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> with a map * : A → A that is an <a href="/facts/Antiautomorphism/nIRWxknY">antiautomorphism</a> and an <a href="/facts/Semigroup_with_involution/aXJEUAmY">involution</a>.
More precisely, * is required to satisfy the following properties:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 

<ul><li>(x + y)* = x* + y*</li>
<li>(x y)* = y* x*</li>
<li>1* = 1</li>
<li>(x*)* = x</li></ul>
for all x, y in A.
This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.
Elements such that x* = x are called <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Archetypical examples of a *-ring are fields of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> and <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a> with <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a> as the involution. One can define a <a href="/facts/Sesquilinear_form/Kp6qUA8v">sesquilinear form</a> over any *-ring.
Also, one can define *-versions of algebraic objects, such as <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> and <a href="/facts/Subring/M0perybt">subring</a>, with the requirement to be *-<a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariant</a>: x ∈ I ⇒ x* ∈ I and so on.
*-rings are unrelated to <a href="/facts/Star_semiring/7cSvWXVQ">star semirings</a> in the theory of computation.

<h3>*-algebra</h3>
A *-algebra A is a *-ring,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> with involution * that is an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a> *-ring R with involution ′, such that (r x)* = r′ x*  ∀r ∈ R, x ∈ A.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
The base *-ring R is often the complex numbers (with ′ acting as complex conjugation).
It follows from the axioms that * on A is <a href="/facts/Conjugate-linear/GTXzWMKc">conjugate-linear</a> in R, meaning

(λ x + μ y)* = λ′ x* + μ′ y*
for λ, μ ∈ R, x, y ∈ A.
A *-homomorphism f : A → B is an <a href="/facts/Algebra_homomorphism/b7YL8aPR">algebra homomorphism</a> that is compatible with the involutions of A and B, i.e.,

<ul><li>f(a*) = f(a)* for all a in A.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li></ul>
<h3>Philosophy of the *-operation</h3>
The *-operation on a *-ring is analogous to <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a> on the complex numbers. The *-operation on a *-algebra is analogous to taking <a href="/facts/Conjugate_transpose/7eBWtqVG">adjoints</a> in complex <a href="/facts/Matrix_algebra/oj5HBr8B">matrix algebras</a>.

<h3>Notation</h3>
The * involution is a <a href="/facts/Unary_operation/kb6CkgsX">unary operation</a> written with a postfixed star glyph centered above or near the <a href="/facts/Mean_line/6Yb8UTha">mean line</a>:

x ↦ x*, or
x ↦ x∗ (<a href="/facts/TeX/HDrl3Yvd">TeX</a>: x^*),
but not as "x∗"; see the <a href="/facts/Asterisk/qwmtPW5V">asterisk</a> article for details.

<h2 id="examples">Examples</h2>
<ul><li>Any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> becomes a *-ring with the trivial (<a href="/facts/Identity_map/9AtsP0LC">identical</a>) involution.</li>
<li>The most familiar example of a *-ring and a *-algebra over <a href="/facts/Real_number/R02gw5Pb">reals</a> is the field of complex numbers C where * is just <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a>.</li>
<li>More generally, a <a href="/facts/Field_extension/L7yIDtyK">field extension</a> made by adjunction of a <a href="/facts/Square_root/AfrzfBdQ">square root</a> (such as the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> √−1) is a *-algebra over the original field, considered as a trivially-*-ring. The * <a href="/facts/Additive_inverse/n8gKUmCf">flips the sign</a> of that square root.</li>
<li>A <a href="/facts/Quadratic_integer/uGsLh5kL">quadratic integer</a> ring (for some D) is a commutative *-ring with the * defined in the similar way; <a href="/facts/Quadratic_field/dUnSDN4e">quadratic fields</a> are *-algebras over appropriate quadratic integer rings.</li>
<li><a href="/facts/Quaternion/T3tnu7mX">Quaternions</a>, <a href="/facts/Split-complex_number/DtavqChA">split-complex numbers</a>, <a href="/facts/Dual_number/DntuRu3F">dual numbers</a>, and possibly other <a href="/facts/Hypercomplex_number/BwKwTBF2">hypercomplex number</a> systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). None of the three is a complex algebra.</li>
<li><a href="/facts/Hurwitz_quaternion/IJ0hdSgN">Hurwitz quaternions</a> form a non-commutative *-ring with the quaternion conjugation.</li>
<li>The <a href="/facts/Matrix_ring/oj5HBr8B">matrix algebra</a> of n × n <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> over R with * given by the <a href="/facts/Transpose/8wmsagGS">transposition</a>.</li>
<li>The matrix algebra of n × n matrices over C with * given by the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>.</li>
<li>Its generalization, the <a href="/facts/Hermitian_adjoint/gN6RHphd">Hermitian adjoint</a> in the algebra of <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operators</a> on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> also defines a *-algebra.</li>
<li>The <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> R[x] over a commutative trivially-*-ring R is a *-algebra over R with P *(x) = P (−x).</li>
<li>If (A, +, ×, *) is simultaneously a *-ring, an <a href="/facts/Algebra_over_a_ring/8T8ulWJb">algebra over a ring</a> R (commutative), and (r x)* = r (x*)  ∀r ∈ R, x ∈ A, then A is a *-algebra over R (where * is trivial).
<ul><li>As a partial case, any *-ring is a *-algebra over <a href="/facts/Integer/PUwwrawx">integers</a>.</li></ul></li>
<li>Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring.</li>
<li>For a commutative *-ring R, its <a href="/facts/Quotient_ring/X2WkXyEw">quotient</a> by any its *-ideal is a *-algebra over R.
<ul><li>For example, any commutative trivially-*-ring is a *-algebra over its <a href="/facts/Dual_number/DntuRu3F">dual numbers ring</a>, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ring.</li>
<li>The same about a commutative ring K and its polynomial ring K[x]: the quotient by x = 0 restores K.</li></ul></li>
<li>In <a href="/facts/Hecke_algebra_of_a_Coxeter_group/GQNLrpaL">Hecke algebra</a>, an involution is important to the <a href="/facts/Kazhdan%25E2%2580%2593Lusztig_polynomial/kC0JqxUM">Kazhdan–Lusztig polynomial</a>.</li>
<li>The <a href="/facts/Endomorphism_ring/K1UgpnJK">endomorphism ring</a> of an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> becomes a *-algebra over the integers, where the involution is given by taking the <a href="/facts/Dual_abelian_variety/xRSRm5KU">dual isogeny</a>. A similar construction works for <a href="/facts/Abelian_variety/dGSDVruG">abelian varieties</a> with a <a href="/facts/Abelian_variety/dGSDVruG">polarization</a>, in which case it is called the <a href="/facts/Rosati_involution/tvTtH9xW">Rosati involution</a> (see Milne's lecture notes on abelian varieties).</li></ul>
<a href="/facts/Hopf_algebra/3QF6g0tT">Involutive Hopf algebras</a> are important examples of *-algebras (with the additional structure of a compatible <a href="/facts/Comultiplication/pvx52xS6">comultiplication</a>); the most familiar example being:

<ul><li>The <a href="/facts/Group_Hopf_algebra/qvwvRqXN">group Hopf algebra</a>: a <a href="/facts/Group_ring/TJlkt5Te">group ring</a>, with involution given by g ↦ g−1.</li></ul>
<h2 id="non-example">Non-Example</h2>
Not every algebra admits an involution:
Regard the 2×2 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> over the complex numbers. Consider the following subalgebra:

A
          
        
        :=
        
          {
          
            
              
                (
                
                  
                    
                      a
                    
                    
                      b
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                  
                
                )
              
            
            :
            a
            ,
            b
            ∈
            
              C
            
          
          }
        
      
    
    {\displaystyle {\mathcal {A}}:=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}:a,b\in \mathbb {C} \right\}}

Any nontrivial antiautomorphism necessarily has the form:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

φ
          
            z
          
        
        
          [
          
            
              (
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                
              
              )
            
          
          ]
        
        =
        
          
            (
            
              
                
                  1
                
                
                  z
                
              
              
                
                  0
                
                
                  0
                
              
            
            )
          
        
        
        
          φ
          
            z
          
        
        
          [
          
            
              (
              
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                
              
              )
            
          
          ]
        
        =
        
          
            (
            
              
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
              
            
            )
          
        
      
    
    {\displaystyle \varphi _{z}\left[{\begin{pmatrix}1&0\\0&0\end{pmatrix}}\right]={\begin{pmatrix}1&z\\0&0\end{pmatrix}}\quad \varphi _{z}\left[{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]={\begin{pmatrix}0&0\\0&0\end{pmatrix}}}

for any complex number 
 
 
 
 z
 ∈
 
 C
 
 
 
 {\displaystyle z\in \mathbb {C} }
 
.
It follows that any nontrivial antiautomorphism fails to be involutive:

φ
          
            z
          
          
            2
          
        
        
          [
          
            
              (
              
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                
              
              )
            
          
          ]
        
        =
        
          
            (
            
              
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
              
            
            )
          
        
        ≠
        
          
            (
            
              
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  0
                
              
            
            )
          
        
      
    
    {\displaystyle \varphi _{z}^{2}\left[{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\neq {\begin{pmatrix}0&1\\0&0\end{pmatrix}}}

Concluding that the subalgebra admits no involution.

<h2 id="additional-structures">Additional structures</h2>
Many properties of the <a href="/facts/Transpose/8wmsagGS">transpose</a> hold for general *-algebras:

<ul><li>The <a href="/facts/Self-adjoint/rKqGtlpy">Hermitian</a> elements form a <a href="/facts/Jordan_algebra/4BBGsQd9">Jordan algebra</a>;</li>
<li>The skew Hermitian elements form a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>;</li>
<li>If 2 is invertible in the *-ring, then the operators ⁠1/2⁠(1 + *) and ⁠1/2⁠(1 − *) are <a href="/facts/Idempotent/khbSpexv">orthogonal idempotents</a>,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of <a href="/facts/Module_(algebra)/jP0LIhFL">modules</a> (<a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. These spaces do not, generally, form associative algebras, because the idempotents are <a href="/facts/Linear_operator/Q1PBKNVV">operators</a>, not elements of the algebra.</li></ul>
<h3>Skew structures</h3>
Given a *-ring, there is also the map −* : x ↦ −x*.
It does not define a *-ring structure (unless the <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> is 2, in which case −* is identical to the original *), as 1 ↦ −1, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where x ↦ x*.
Elements fixed by this map (i.e., such that a = −a*) are called skew Hermitian.
For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Semigroup_with_involution/aXJEUAmY">Semigroup with involution</a></li>
<li><a href="/facts/B*-algebra/B3tnGHIo">B*-algebra</a></li>
<li><a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a></li>
<li><a href="/facts/Dagger_category/qE2Vr7B9">Dagger category</a></li>
<li><a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebra</a></li>
<li><a href="/facts/Baer_ring/2hF5faYW">Baer ring</a></li>
<li><a href="/facts/Operator_algebra/4fycDNQ1">Operator algebra</a></li>
<li><a href="/facts/Conjugate_(algebra)/6jbUaXd3">Conjugate (algebra)</a></li>
<li><a href="/facts/Cayley%25E2%2580%2593Dickson_construction/aDmlkB9m">Cayley–Dickson construction</a></li>
<li><a href="/facts/Composition_algebra/Y230strm">Composition algebra</a></li></ul>
<h2 id="notes">Notes</h2>

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

<ol>
<li id="fn:1">In this context, involution is taken to mean an involutory antiautomorphism, also known as an anti-involution. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weisstein, Eric W. (2015). "C-Star Algebra". Wolfram MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 26 March 2015. Retrieved 27 January 2015. <a href="/wiki/John_Baez" target="_blank">/wiki/John_Baez</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only. <a href="/wiki/Multiplicative_identity" target="_blank">/wiki/Multiplicative_identity</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">star-algebra at the nLab <a href="https://ncatlab.org/nlab/show/star-algebra" target="_blank">https://ncatlab.org/nlab/show/star-algebra</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 26 March 2015. Retrieved 27 January 2015. <a href="/wiki/John_Baez" target="_blank">/wiki/John_Baez</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Winker, S. K.; Wos, L.; Lusk, E. L. (1981). "Semigroups, Antiautomorphisms, and Involutions: A Computer Solution to an Open Problem, I". Mathematics of Computation. 37 (156): 533–545. doi:10.2307/2007445. ISSN 0025-5718. <a href="https://www.jstor.org/stable/2007445" target="_blank">https://www.jstor.org/stable/2007445</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 26 March 2015. Retrieved 27 January 2015. <a href="/wiki/John_Baez" target="_blank">/wiki/John_Baez</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

*-algebra open-in-new

*-algebra