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.,
for all in.
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.
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
Any commutative ring becomes a *-ring with the trivial involution.
The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers where * is just complex conjugation.
More generally, a field extension made by adjunction of a square root is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.
Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *-rings and *-algebras over reals. Note that neither of the three is a complex algebra.
Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation.
Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
The polynomial ring over a commutative trivially-*-ring is a *-algebra over with.
If is simultaneously a *-ring, an algebra over a ring , and, then is a *-algebra over .
* As a partial case, any *-ring is a *-algebra over integers.
Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring.
For a commutative *-ring, its quotient by any its *-ideal is a *-algebra over.
* For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by makes the original ring.
* The same about a commutative ring and its polynomial ring : the quotient by restores.
Involutive Hopf algebras are important examples of *-algebras ; the most familiar example being:
The group Hopf algebra: a group ring, with involution given by.
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:
If 2 is invertible in the *-ring, then and are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules of symmetric and anti-symmetric 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. 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.
