*-algebra


In mathematics, and more specifically in abstract algebra, a *-algebra is a mathematical structure consisting of two involutive rings and, where is commutative and has the structure of an associative algebra over. 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.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map that is an antiautomorphism and an involution.
More precisely, is required to satisfy the following properties:
for all in.
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 is also a multiplicative identity, and identities are unique.
Elements such that 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: and so on.

*-algebra

A *-algebra is a *-ring, with involution * that is an associative algebra over a commutative *-ring with involution, such that.
The base *-ring is often the complex numbers.
It follows from the axioms that * on is conjugate-linear in, meaning
for.
A *-homomorphism is an algebra homomorphism that is compatible with the involutions of and, i.e.,
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.

Notation

The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:
but not as ""; see the asterisk article for details.

Examples

Involutive Hopf algebras are important examples of *-algebras ; the most familiar example being:
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:
Given a *-ring, there is also the map.
It does not define a *-ring structure, as, neither is it antimultiplicative, but it satisfies the other axioms and hence is quite similar to *-algebra where.
Elements fixed by this map 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.