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.

Look up * or star in Wiktionary, the free dictionary.



In mathematics, a *-ring is a ring with a map * : AA that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:[1]

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.[2]

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: xIx* ∈ I and so on.


A *-algebra A is a *-ring,[lower-alpha 1] with involution * that is an associative algebra over a commutative *-ring R with involution , such that (r x)* = r′x* rR, xA.[3]

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  rR, xA

the * on A is conjugate-linear in R, meaning

(λ x + μy)* = λx* + μy*

for λ, μR, x, yA.

A *-homomorphism f : AB 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:

xx*, or
xx (TeX: x^*),

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:

Additional structures

Many properties of the transpose hold for general *-algebras:

Skew structures

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 xx*.

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.

See also


  1. Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only.


  1. Weisstein, Eric W. (2015). "C-Star Algebra". Wolfram MathWorld.
  2. 1 2 3 Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 25 March 2015. Retrieved 27 January 2015.
  3. star-algebra in nLab
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