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, yinA.
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 Aisconjugate-linearinR, 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.
Aquadratic 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.
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