*-algebra

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 $* : A \to A$ 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.

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