In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-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.
[a] Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra 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:[1] for all x, y in A.
Elements such that x* = x are called self-adjoint.
[2] Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution.
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.
*-rings are unrelated to star semirings in the theory of computation.
It follows from the axioms that * on A is conjugate-linear in R, meaning 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., 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 * involution is a unary operation written with a postfixed star glyph centered above or near the mean line: 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 2×2 matrices over the complex numbers.
Any nontrivial antiautomorphism necessarily has the form:[4]
It follows that any nontrivial antiautomorphism fails to be involutive:
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.