Involutive algebra

In, and more specifically in , a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings $R$ and $A$, where $R$ is commutative and $A$ has the structure of an over $R$. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the and,  over the complex numbers and , and s over a  and s. However, it may happen that an algebra admits no involution at all.

*-ring
In, a *-ring is a with a map $
 * : A → A$ that is an  and an.

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. Note that the third axiom is actually redundant, because the second and fourth axioms imply $(x y)* = y* x*$ is also a multiplicative identity, and are unique.

Elements such that $1* = 1$ are called .

Archetypical examples of a *-ring are fields of s and s with as the involution. One can define a over any *-ring.

Also, one can define *-versions of algebraic objects, such as and, with the requirement to be *-: $(x*)* = x$ and so on.

*-algebra
A *-algebra $A$ is a *-ring, with involution * that is an over a  *-ring $R$ with involution $′$, such that $x, y$.

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

It follows from the axioms that * on $A$ is in $R$, meaning

for $1*$.

A *-homomorphism $x* = x$ is an 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 on the complex numbers. The *-operation on a *-algebra is analogous to taking in $(r x)* = r′ x* ∀r ∈ R, x ∈ A$.

Notation
The * involution is a written with a postfixed star glyph centered above or near the :
 * $(&lambda; x + &mu; y)* = &lambda;′ x* + &mu;′ y*$, or

but not as "$&lambda;, &mu; ∈ R, x, y ∈ A$"; see the article for details.

Examples
are important examples of *-algebras (with the additional structure of a compatible ); the most familiar example being:
 * Any becomes a *-ring with the trivial  involution.
 * The most familiar example of a *-ring and a *-algebra over is the field of complex numbers $f : A → B$ where * is just.
 * More generally, a made by adjunction of a  (such as the  √−1) is a *-algebra over the original field, considered as a trivially-*-ring. The *  of that square root.
 * A ring (for some $D$) is a commutative *-ring with the * defined in the similar way; s are *-algebras over appropriate quadratic integer rings.
 * s, s, s, and possibly other systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). Note that neither of the three is a complex algebra.
 * s form a non-commutative *-ring with the quaternion conjugation.
 * The of $f(a*) = f(a)*$ over R with * given by the.
 * The matrix algebra of $(C)$matrices over C with * given by the.
 * Its generalization, the in the algebra of s on a  also defines a *-algebra.
 * The $x ↦ x*$ over a commutative trivially-*-ring $R$ is a *-algebra over $R$ with $x ↦ x^{∗}$.
 * If $x∗$ is simultaneously a *-ring, an $R$ (commutative), and $C$, then $A$ is a *-algebra over $R$ (where * is trivial).
 * As a partial case, any *-ring is a *-algebra over s.
 * Any commutative *-ring is a *-algebra over itself and, more generally, over any its.
 * For a commutative *-ring $R$, its by any its  is a *-algebra over $R$.
 * For example, any commutative trivially-*-ring is a *-algebra over its, 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 $n × n$: the quotient by $R[x]$ restores $K$.
 * In, an involution is important to the.
 * The of an  becomes a *-algebra over the integers, where the involution is given by taking the . A similar construction works for  with a, in which case it is called the  (see Milne's lecture notes on abelian varieties).
 * The : a, with involution given by $P *(x) = P (−x)$.

Non-Example
Not every algebra admits an involution:

Regard the 2x2 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 idempotent:

$$\varphi_z^2\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right]=\begin{pmatrix}0&0\\0&0\end{pmatrix}\neq\begin{pmatrix}0&1\\0&0\end{pmatrix}$$

Concluding that the subalgebra admits no involution.

Additional structures
Many properties of the hold for general *-algebras:
 * The Hermitian elements form a ;
 * The skew Hermitian elements form a ;
 * If 2 is invertible in the *-ring, then $1⁄2$$(A, +, ×, *)$ and $1⁄2$$(r x)* = r (x*) ∀r ∈ R, x ∈ A$ are, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of (s 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 , not elements of the algebra.

Skew structures
Given a *-ring, there is also the map $ε = 0$. It does not define a *-ring structure (unless the is 2, in which case −* is identical to the original *), as $K[x]$, neither is it antimultiplicative,  but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where $x = 0$.

Elements fixed by this map (i.e., such that $g ↦ g^{−1}$) 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.