In mathematics, and more specifically in abstract algebra, 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 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 at all.

## Contents

## *-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**

*x**)* =

*x*

for all *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 1* is also a multiplicative identity, and identities are unique.

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.

## *-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 usually the complex numbers (with ′ acting as complex conjugation) and is commutative with A such that A is both left and right algebra.

Since R is central in A, that is,

*rx*=

*xr*∀

*r*∈

*R*,

*x*∈

*A*

the * 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 GL_{n}(**C**).

## 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

**C**where * is just complex conjugation.

*n*×

*n*matrices over

**R**with * given by the transposition.

*n*×

*n*matrices over

**C**with * given by the conjugate transpose.

*R*[

*x*] over a commutative trivially-*-ring R is a *-algebra over R with

*P**(

*x*) =

*P*(−

*x*).

*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).

*non-trivial**, because the quotient by ε = 0 makes the original ring.

*K*[

*x*]: the quotient by

*x*= 0 restores K.

Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:

*g*↦

*g*

^{−1}.

## Non-Example

Not every algebra admits an involution:

Regard the 2x2 matrices over the complex numbers.

Consider the following subalgebra:

Any nontrivial antiautomorphism necessarily has the form:

for any complex number

It follows that any nontrivial antiautomorphism fails to be idempotent:

Concluding that the subalgebra admits no involution.

## Additional structures

Many properties of the transpose hold for general *-algebras:

*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.