*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive 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.
Definitions.
*-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 (with * acting as complex conjugation).
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.,
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:
but not as ""; 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:
Non-Example.
Not every algebra admits an involution:
Regard the 2x2 matrices over the complex numbers.<br>
Consider the following subalgebra:
formula_1
Any nontrivial antiautomorphism necessarily has the form:
formula_2
for any complex number formula_3.<br>
It follows that any nontrivial antiautomorphism fails to be idempotent:
formula_4
Concluding that the subalgebra admits no involution.
Additional structures.
Many properties of the transpose hold for general *-algebras:
Skew structures.
Given a *-ring, there is also the map .
It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as , neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where .
Elements fixed by this map (i.e., such that ) 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.