*-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.

*-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: for all $(x + y)* = x* + y*$ 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 $(x y)* = y* x*$ 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: $1* = 1$ and so on.

&ast;-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 $(x*)* = x$.

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

for $x, y$.

A *-homomorphism $x* = x$ is an algebra homomorphism that is compatible with the involutions of $A$ and $B$, i.e.,
 * $x ∈ I ⇒ x* ∈ I$ 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:
 * $(r x)* = r x* ∀r ∈ R, x ∈ A$, or
 * $(λ x + μ y)* = λ x* + μ y*$ (TeX: ),

but not as "$λ, μ ∈ R, x, y ∈ A$"; see the asterisk article for details.

Examples
Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:
 * 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 $f : A → B$ 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 $f(a*) = f(a)*$matrices over R with * given by the transposition.
 * The matrix algebra of $x ↦ x*$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 $x ↦ x^{∗}$ over a commutative trivially-*-ring $R$ is a *-algebra over $R$ with $x∗$.
 * If $C$ is simultaneously a *-ring, an algebra over a ring $R$ (commutative), and $n × n$, 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 $n × n$ makes the original ring.
 * The same about a commutative ring $K$ and its polynomial ring $R[x]$: the quotient by $P *(x) = P (−x)$ 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).
 * The group Hopf algebra: a group ring, with involution given by $(A, +, ×, *)$.

Non-Example
Not every algebra admits an involution:

Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra: $$\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}$$

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 $$z\in\Complex$$.

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 $(r x)* = r (x*) ∀r ∈ R, x ∈ A$ and $ε = 0$ 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 $K[x]$. It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as $x = 0$, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where $g ↦ g^{−1}$.

Elements fixed by this map (i.e., such that $1⁄2(1 + *)$) 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.