*-algebra

{{Short description|Mathematical structure in abstract algebra}} {{Algebraic structures}} In mathematics, and more specifically in 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 over {{mvar|R}}. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution.{{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, a '''*-ring''' is a ring with a map {{math|* : ''A'' → ''A''}} that is an antiautomorphism and an 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''.<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 numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring.

{{anchor|*-objects}}Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: {{math|''x'' ∈ ''I'' ⇒ ''x''* ∈ ''I''}} and so on.

&ast;-rings are unrelated to star semirings 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, i.e. a *-algebra is allowed to be a *-rng only.}} with involution * that is an associative algebra over a commutative *-ring {{mvar|R}} with involution {{mvar|{{prime}}}}, such that {{math|1=(''r x'')* = ''r{{prime}}'' ''x''*  ∀''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 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 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 on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in complex matrix algebras.

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

==Examples== * Any commutative ring becomes a *-ring with the trivial (identical) involution. * The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers {{math|'''C'''}} where * is just complex conjugation. * More generally, a field extension made by adjunction of a square root (such as the imaginary unit {{sqrt|−1}}) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root. * A quadratic integer ring (for some {{mvar|D}}) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings. * Quaternions, split-complex numbers, dual numbers, and possibly other 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 quaternions form a non-commutative *-ring with the quaternion conjugation. * The matrix algebra of {{math|''n'' × ''n'' }}matrices over '''R''' with * given by the transposition. * The matrix algebra of {{math|''n'' × ''n'' }}matrices over '''C''' with * given by the conjugate transpose. * Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra. * The 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 {{mvar|R}} (commutative), and {{math|1=(''r x'')* = ''r'' (''x''*)  ∀''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 integers. * Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring. * For a commutative *-ring {{mvar|R}}, its quotient by any its *-ideal is a *-algebra over {{mvar|R}}. ** For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with ''non-trivial'' *, because the quotient by {{math|1=ε = 0}} makes the original 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, an involution is important to the Kazhdan–Lusztig polynomial. * The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties). Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being: * The group Hopf algebra: a 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 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 hold for general *-algebras: * The Hermitian elements form a Jordan algebra; * The skew Hermitian elements form a 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,<ref name=":0" /> called ''symmetrizing'' and ''anti-symmetrizing'', so the algebra decomposes as a direct sum of modules (vector spaces 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, 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 is 2, in which case −* is identical to the original *), as {{math|1 ↦ −1}}, 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.

==Notes== {{noteslist}}

==References== {{reflist}}

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