*-algebra

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.

Definitions

*-ring

In mathematics, a *-ring is a ring A 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.