More precisely, * is required to satisfy the following properties:[1]
(x + y)* = x* + y*
(x y)* = y* x*
1* = 1
(x*)* = x
for all x, y in A.
This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.
A *-algebraA is a *-ring,[b] with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (r x)* = r′x* ∀r ∈ R, x ∈ A.[3]
The base *-ring R is often the complex numbers (with ′ acting as complex conjugation).
It follows from the axioms that * on A is conjugate-linear in R, meaning
(λ x + μy)* = λ′x* + μ′y*
for λ, μ ∈ R, x, y ∈ A.
A *-homomorphismf : A → B is an algebra homomorphism that is compatible with the involutions of A and B, 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 complex matrix algebras.
The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers C where * is just complex conjugation.
More generally, a field extension made by adjunction of a square root (such as the imaginary unit√−1) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.
A quadratic integer ring (for some D) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings.
The polynomial ringR[x] over a commutative trivially-*-ring R is a *-algebra over R with P *(x) = P (−x).
If (A, +, ×, *) is simultaneously a *-ring, an algebra over a ringR (commutative), and (r x)* = r (x*) ∀r ∈ R, x ∈ A, then A is a *-algebra over R (where * is trivial).
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 R, its quotient by any its *-ideal is a *-algebra over R.
For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ring.
The same about a commutative ring K and its polynomial 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:
If 2 is invertible in the *-ring, then the operators 1/2(1 + *) and 1/2(1 − *) are orthogonal idempotents,[2] 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.
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.
^ abcBaez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 26 March 2015. Retrieved 27 January 2015.