In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. 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.
In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.
More precisely, * is required to satisfy the following properties:
for all x, y in A.
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 1* is also a multiplicative identity, and identities are unique.
Elements such that x* = x 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: x ∈ I ⇒ x* ∈ I and so on.
A *-algebra A is a *-ring, with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (r x)* = r′ x* ∀r ∈ R, x ∈ A.
The base *-ring R is usually the complex numbers (with ′ acting as complex conjugation) and is commutative with A such that A is both left and right algebra.
Since R is central in A, that is,rx = xr ∀r ∈ R, x ∈ A
the * on A is conjugate-linear in R, meaning(λ x + μ y)* = λ′ x* + μ′ y*
for λ, μ ∈ R, x, y ∈ A.
A *-homomorphism f : A → B is an algebra homomorphism that is compatible with the involutions of A and B, 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 GLn(C).
The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:x ↦ x*, or x ↦ x∗ (TeX:
but not as "x∗"; see the asterisk article for details.
Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); 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.
Many properties of the transpose hold for general *-algebras:
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.