# *-algebra

{{ safesubst:#invoke:Unsubst||\$N=Refimprove |date=__DATE__ |\$B= {{#invoke:Message box|ambox}} }} In mathematics, and more specificially in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings Template:Mvar and Template:Mvar, where Template:Mvar is commutative and Template:Mvar has the structure of an associative algebra over Template:Mvar. 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. Template:Sister

## Terminology

### *-ring

In mathematics, a *-ring is an associative ring with a map * : AA which 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 Template:Mvar.

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.

{{safesubst:#invoke:anchor|main}}Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: xIx* ∈ I and so on.

### *-algebra

A *-algebra Template:Mvar is a *-ring, with involution * that is an associative algebra over a commutative *-ring Template:Mvar with involution Template:Mvar, such that (r x)* = r′x*  ∀rR, xA.

The base *-ring is usually the complex numbers (with Template:Mvar acting as complex conjugation) and commutes with Template:Mvar.Template:Clarify

Since Template:Mvar is central,Template:Clarify the * on Template:Mvar is conjugate-linear in Template:Mvar, meaning

(λ x + μy)* = λ′x* + μ′y*

for λ, μR, x, yA.

A *-homomorphism f : AB is an algebra homomorphism that is compatible with the involutions of Template:Mvar and Template:Mvar, i.e.,

### *-operation

A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarlyTemplate:Vague to taking adjoints in GLn(C).

### Notation

The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:

xx*, or
xx (TeX: `x^*`),

but not as "x"; 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:

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 {{ safesubst:#invoke:Unsubst||\$B=1/2}}(1 + *) and {{ safesubst:#invoke:Unsubst||\$B=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.Template:Vague

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.