# *-algebra

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.

## Definitions
### *-ring

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:
* (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.
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.

### *-algebra

A *-algebra A is a *-ring, with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (r x)* = r′ x* ∀r ∈ R, x ∈ A.
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.

### 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:
    x ↦ x*, or
    x ↦ x$^∗$ (TeX: x^*),
but not as "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 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). None 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.
* 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$^{−1}$.

## Non-Example

Not every algebra admits an involution:
Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra: ${\mathcal {A}}:=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}:a,b\in \mathbb {C} \right\}$
Any nontrivial antiautomorphism necessarily has the form: $\varphi _{z}\left[{\begin{pmatrix}1&0\\0&0\end{pmatrix}}\right]={\begin{pmatrix}1&z\\0&0\end{pmatrix}}\quad \varphi _{z}\left[{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]={\begin{pmatrix}0&0\\0&0\end{pmatrix}}$ for any complex number $z\in \mathbb {C}$.
It follows that any nontrivial antiautomorphism fails to be involutive: $\varphi _{z}^{2}\left[{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\neq {\begin{pmatrix}0&1\\0&0\end{pmatrix}}$
Concluding that the subalgebra admits no involution.

## 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 ⁠1/2⁠(1 + *) and ⁠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.

### Skew structures

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.