# *-algebra

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{{Short description|Mathematical structure in abstract algebra}}
{{Algebraic structures}}
In [mathematics](/source/mathematics), and more specifically in [abstract algebra](/source/abstract_algebra), a '''*-algebra''' (or '''involutive algebra'''; read as "star-algebra") is a mathematical structure consisting of two '''involutive rings''' {{mvar|R}} and {{mvar|A}}, where {{mvar|R}} is commutative and {{mvar|A}} has the structure of an [associative algebra](/source/associative_algebra) over {{mvar|R}}. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the [complex numbers](/source/complex_numbers) and [complex conjugation](/source/complex_conjugation), [matrices](/source/Matrix_(mathematics)) over the complex numbers and [conjugate transpose](/source/conjugate_transpose), and [linear operator](/source/linear_operator)s over a [Hilbert space](/source/Hilbert_space) and [Hermitian adjoint](/source/Hermitian_adjoint)s.
However, it may happen that an algebra admits no [involution](/source/involution_(mathematics)).{{efn|In this context, ''involution'' is taken to mean an involutory antiautomorphism, also known as an ''anti-involution''.}}
{{wiktionary|*|star}}

== Definitions ==

===*-ring===
{{Ring theory sidebar}}
In [mathematics](/source/mathematics), a '''*-ring''' is a [ring](/source/ring_(mathematics)) with a map {{math|* : ''A'' → ''A''}} that is an [antiautomorphism](/source/antiautomorphism) and an [involution](/source/Semigroup_with_involution).

More precisely, {{math|*}} is required to satisfy the following properties:<ref>{{Cite web |url=http://mathworld.wolfram.com/C-Star-Algebra.html |title=C-Star Algebra |website = Wolfram MathWorld |date=2015 |first=Eric W. |last=Weisstein|authorlink = Eric W. Weisstein}}</ref> 
* {{math|size=120%|1=(''x'' + ''y'')* = ''x''* + ''y''*}}
* {{math|size=120%|1=(''x y'')* = ''y''* ''x''*}}
* {{math|size=120%|1=1* = 1}}
* {{math|size=120%|1=(''x''*)* = ''x''}}
for all {{math|''x'', ''y''}} in {{mvar|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 {{math|1=''x''* = ''x''}} are called ''[self-adjoint](/source/self-adjoint)''.<ref name=":0">{{Cite web|url=http://math.ucr.edu/home/baez/octonions/node5.html |title=Octonions |date=2015 |accessdate=27 January 2015 |website=Department of Mathematics |publisher=University of California, Riverside |last=Baez |first=John |author-link = John Baez|archiveurl=https://web.archive.org/web/20150326133405/http://math.ucr.edu/home/baez/octonions/node5.html |archivedate=26 March 2015 |url-status=live |df= }}</ref>

Archetypical examples of a *-ring are fields of [complex number](/source/complex_number)s and [algebraic number](/source/algebraic_number)s with [complex conjugation](/source/complex_conjugation) as the involution.  One can define a [sesquilinear form](/source/sesquilinear_form) over any *-ring.

{{anchor|*-objects}}Also, one can define *-versions of algebraic objects, such as [ideal](/source/ideal_(ring_theory)) and [subring](/source/subring), with the requirement to be *-[invariant](/source/invariant_(mathematics)): {{math|''x'' ∈ ''I'' ⇒ ''x''* ∈ ''I''}} and so on.

&ast;-rings are unrelated to [star semirings](/source/Star_semiring) in the theory of computation.

===*-algebra===
A '''*-algebra''' {{mvar|A}} is a *-ring,{{efn|Most definitions do not require a *-algebra to have the [unity](/source/multiplicative_identity), i.e. a *-algebra is allowed to be a *-[rng](/source/rng_(algebra)) only.}} with involution * that is an [associative algebra](/source/associative_algebra) over a [commutative](/source/commutative_ring) *-ring {{mvar|R}} with involution {{mvar|{{prime}}}}, such that {{math|1=(''r x'')* = ''r{{prime}}'' ''x''*&nbsp; ∀''r'' ∈ ''R'', ''x'' ∈ ''A''}}.<ref>{{nlab|id=star-algebra}}</ref>

The base *-ring {{mvar|R}} is often the complex numbers (with {{mvar|{{prime}}}} acting as complex conjugation).

It follows from the axioms that * on {{mvar|A}} is [conjugate-linear](/source/conjugate-linear) in {{mvar|R}}, meaning
:{{math|size=120%|1=(''λ x'' + ''μ'' ''y'')* = ''λ{{prime}}'' ''x''* + ''μ{{prime}}'' ''y''*}}
for {{math|''λ'', ''μ'' ∈ ''R'', ''x'', ''y'' ∈ ''A''}}.

A '''*-homomorphism''' {{math|''f'' : ''A'' → ''B''}} is an [algebra homomorphism](/source/algebra_homomorphism) that is compatible with the involutions of {{mvar|A}} and {{mvar|B}}, i.e.,
* {{math|size=120%|1=''f''(''a''*) = ''f''(''a'')*}} for all {{mvar|a}} in {{mvar|A}}.<ref name=":0" />

===Philosophy of the *-operation===
The *-operation on a *-ring is analogous to [complex conjugation](/source/complex_conjugation) on the complex numbers. The *-operation on a *-algebra is analogous to taking [adjoints](/source/conjugate_transpose) in complex [matrix algebra](/source/matrix_algebra)s.

===Notation===
The * involution is a [unary operation](/source/unary_operation) written with a postfixed star glyph centered above or near the [mean line](/source/mean_line):
: {{math|size=120%|''x'' ↦ ''x''*}}, or 
: {{math|size=120%|''x'' ↦ ''x''<sup>∗</sup>}} ([TeX](/source/TeX): <code>x^*</code>),
but not as "{{math|''x''∗}}"; see the [asterisk](/source/asterisk) article for details.

==Examples==
* Any [commutative ring](/source/commutative_ring) becomes a *-ring with the trivial ([identical](/source/identity_map)) involution. 
* The most familiar example of a *-ring and a *-algebra over [reals](/source/real_number) is the field of complex numbers {{math|'''C'''}} where * is just [complex conjugation](/source/complex_conjugation).
* More generally, a [field extension](/source/field_extension) made by adjunction of a [square root](/source/square_root) (such as the [imaginary unit](/source/imaginary_unit) {{sqrt|−1}}) is a *-algebra over the original field, considered as a trivially-*-ring. The * [flips the sign](/source/additive_inverse) of that square root.
* A [quadratic integer](/source/quadratic_integer) ring (for some {{mvar|D}}) is a commutative *-ring with the * defined in the similar way; [quadratic field](/source/quadratic_field)s are *-algebras over appropriate quadratic integer rings.
* [Quaternion](/source/Quaternion)s, [split-complex number](/source/split-complex_number)s, [dual number](/source/dual_number)s, and possibly other [hypercomplex number](/source/hypercomplex_number) systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). None of the three is a complex algebra.
* [Hurwitz quaternion](/source/Hurwitz_quaternion)s form a non-commutative *-ring with the quaternion conjugation.
* The [matrix algebra](/source/Matrix_ring) of {{math|''n'' × ''n'' }}[matrices](/source/matrix_(mathematics)) over '''R''' with * given by the [transposition](/source/transpose).
* The matrix algebra of {{math|''n'' × ''n'' }}matrices over '''C''' with * given by the [conjugate transpose](/source/conjugate_transpose).
* Its generalization, the [Hermitian adjoint](/source/Hermitian_adjoint) in the algebra of [bounded linear operator](/source/bounded_linear_operator)s on a [Hilbert space](/source/Hilbert_space) also defines a *-algebra.
* The [polynomial ring](/source/polynomial_ring) {{math|''R''[''x'']}} over a commutative trivially-*-ring {{mvar|R}} is a *-algebra over {{mvar|R}} with {{math|1=''P ''*(''x'') = ''P ''(−''x'')}}.
* If {{math|(''A'', +, ×, *)}} is simultaneously a *-ring, an [algebra over a ring](/source/algebra_over_a_ring) {{mvar|R}} (commutative), and {{math|1=(''r x'')* = ''r'' (''x''*)&nbsp; ∀''r'' ∈ ''R'', ''x'' ∈ ''A''}}, then {{mvar|A}} is a *-algebra over {{mvar|R}} (where * is trivial).
** As a partial case, any *-ring is a *-algebra over [integer](/source/integer)s.
* Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring.
* For a commutative *-ring {{mvar|R}}, its [quotient](/source/quotient_ring) by any its *-ideal is a *-algebra over {{mvar|R}}.
** For example, any commutative trivially-*-ring is a *-algebra over its [dual numbers ring](/source/Dual_number), a *-ring with ''non-trivial'' *, because the quotient by {{math|1=ε = 0}} makes the original ring.<!-- is the same true for a non-trivially-* ring? -->
** The same about a commutative ring {{mvar|K}} and its polynomial ring {{math|''K''[''x'']}}: the quotient by {{math|1=''x'' = 0}} restores {{mvar|K}}.
* In [Hecke algebra](/source/Hecke_algebra_of_a_Coxeter_group), an involution is important to the [Kazhdan–Lusztig polynomial](/source/Kazhdan%E2%80%93Lusztig_polynomial).
* The [endomorphism ring](/source/endomorphism_ring) of an [elliptic curve](/source/elliptic_curve) becomes a *-algebra over the integers, where the involution is given by taking the [dual isogeny](/source/dual_abelian_variety). A similar construction works for [abelian varieties](/source/abelian_variety) with a [polarization](/source/abelian_variety), in which case it is called the [Rosati involution](/source/Rosati_involution) (see Milne's lecture notes on abelian varieties).
[Involutive Hopf algebras](/source/Hopf_algebra) are important examples of *-algebras (with the additional structure of a compatible [comultiplication](/source/comultiplication)); the most familiar example being:
* The [group Hopf algebra](/source/group_Hopf_algebra): a [group ring](/source/group_ring), with involution given by {{math|''g'' ↦ ''g''<sup>−1</sup>}}.

==Non-Example==

Not every algebra admits an involution:

Regard the 2×2 [matrices](/source/Matrix_(mathematics)) over the complex numbers. Consider the following subalgebra:
<math display="block">\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}</math>

Any nontrivial antiautomorphism necessarily has the form:<ref>{{Cite journal |last=Winker |first=S. K. |last2=Wos |first2=L. |last3=Lusk |first3=E. L. |date=1981 |title=Semigroups, Antiautomorphisms, and Involutions: A Computer Solution to an Open Problem, I |url=https://www.jstor.org/stable/2007445 |journal=Mathematics of Computation |volume=37 |issue=156 |pages=533–545 |doi=10.2307/2007445 |issn=0025-5718|url-access=subscription }}</ref>
<math display="block">\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}</math>
for any complex number <math>z\in\Complex</math>.

It follows that any nontrivial antiautomorphism fails to be involutive:
<math display="block">\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}</math>

Concluding that the subalgebra admits no involution.

==Additional structures==
Many properties of the [transpose](/source/transpose) hold for general *-algebras:
* The [Hermitian](/source/self-adjoint) elements form a [Jordan algebra](/source/Jordan_algebra);
* The skew Hermitian elements form a [Lie algebra](/source/Lie_algebra);
* If 2 is invertible in the *-ring, then the operators {{math|{{sfrac|1|2}}(1 + *)}} and {{math|{{sfrac|1|2}}(1 − *)}} are [orthogonal idempotents](/source/idempotent),<ref name=":0" /> called ''symmetrizing'' and ''anti-symmetrizing'', so the algebra decomposes as a direct sum of [modules](/source/module_(algebra)) ([vector space](/source/vector_space)s 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 [operators](/source/linear_operator), not elements of the algebra.

===Skew structures===
Given a *-ring, there is also the map {{math|−* : ''x'' ↦ −''x''*}}.
It does not define a *-ring structure (unless the [characteristic](/source/characteristic_(algebra)) is 2, in which case −* is identical to the original *), as {{math|1 ↦ −1}}<!-- (so * is not a ring homomorphism)  /irrelevant -->, neither is it antimultiplicative,  but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where {{math|size=120%|''x'' ↦ ''x''*}}.

Elements fixed by this map (i.e., such that {{math|1=''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.

==See also==
*[Semigroup with involution](/source/Semigroup_with_involution)
*[B*-algebra](/source/B*-algebra)
*[C*-algebra](/source/C*-algebra)
*[Dagger category](/source/Dagger_category)
*[von Neumann algebra](/source/von_Neumann_algebra)
*[Baer ring](/source/Baer_ring)
*[Operator algebra](/source/Operator_algebra)
*[Conjugate (algebra)](/source/Conjugate_(algebra))
*[Cayley–Dickson construction](/source/Cayley%E2%80%93Dickson_construction)
*[Composition algebra](/source/Composition_algebra)

==Notes==
{{noteslist}}

==References==
{{reflist}}

{{Spectral theory}}

{{DEFAULTSORT:-algebra}}
Category:Algebras
Category:Ring theory

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Adapted from the Wikipedia article [*-algebra](https://en.wikipedia.org/wiki/*-algebra) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/*-algebra?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.