*-algebra

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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:{{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}}

• {{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.{{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= }}

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.

*-rings are unrelated to star semirings in the theory of computation.

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}}.{{nlab|id=star-algebra}}

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}}.

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.

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^∗}} (TeX: x^*),

but not as "{{math|x∗}}"; see the asterisk article for details.

• 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⁻¹}}.

Not every algebra admits an involution:

Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra:

Any nontrivial antiautomorphism necessarily has the form:{{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}}

for any complex number $z\backslash in\backslash Complex$.

It follows that any nontrivial antiautomorphism fails to be involutive:

Concluding that the subalgebra admits no involution.

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, 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.

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.

• Semigroup with involution
• B*-algebra
• C*-algebra
• Dagger category
• von Neumann algebra
• Baer ring
• Operator algebra
• Conjugate (algebra)
• Cayley–Dickson construction
• Composition algebra

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Category:Algebras

Category:Ring theory