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Concept
*-algebra
Summary
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.
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:
(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.
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.
*-rings are unrelated to star semirings in the theory of computation.
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 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 + 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.,
f(a) = f(a) for all a in A.
The *-operation on a *-ring is analogous to complex conjugation on the complex numbers.
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