Algebraic structures |
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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** R and A, where R is commutative and A has the structure of an associative algebra over 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.^{[a]}

Algebraic structure → Ring theory Ring theory |
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In mathematics, a ***-ring** is a ring with a map * : *A* → *A* that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:^{[1]}

- (
*x*+*y*)* =*x** +*y** - (
*x y*)* =*y***x** - 1* = 1
- (
*x**)* =*x*

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 *self-adjoint*.^{[2]}

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.

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

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

A ***-algebra** A is a *-ring,^{[b]} with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (*r x*)* = *r′* *x** ∀*r* ∈ *R*, *x* ∈ *A*.^{[3]}

The base *-ring R is often the complex numbers (with ′ acting as complex conjugation).

It follows from the axioms that * on A is conjugate-linear 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 algebra homomorphism that is compatible with the involutions of A and B, i.e.,

*f*(*a**) =*f*(*a*)* for all a in A.^{[2]}

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:

*x*↦*x**, or*x*↦*x*^{∗}(TeX:`x^*`

),

but not as "*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
**C**where * is just complex conjugation. - More generally, a field extension made by adjunction of a square root (such as the imaginary unit √−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 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). Neither of the three is a complex algebra.
- Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation.
- The matrix algebra of
*n*×*n*matrices over**R**with * given by the transposition. - The matrix algebra of
*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
*R*[*x*] over a commutative trivially-*-ring R is a *-algebra over R with*P**(*x*) =*P*(−*x*). - If (
*A*, +, ×, *) is simultaneously a *-ring, an algebra over a ring R (commutative), and (*r x*)* =*r*(*x**) ∀*r*∈*R*,*x*∈*A*, then A is a *-algebra over 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 R, its quotient by any its *-ideal is a *-algebra over R.
- For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with
*non-trivial**, because the quotient by ε = 0 makes the original ring. - The same about a commutative ring K and its polynomial ring
*K*[*x*]: the quotient by*x*= 0 restores K.

- For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with
- 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
*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:^{[4]}

for any complex number .

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 1/2(1 + *) and 1/2(1 − *) are orthogonal idempotents,
^{[2]}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 −* : *x* ↦ −*x**.
It does not define a *-ring structure (unless the characteristic 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.