*-algebra

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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.

Terminology[edit]

*-ring[edit]

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:

  • (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. 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.

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.

*-algebra[edit]

A *-algebra A is a *-ring,[1] with involution * that is an associative algebra over a commutative *-ring R with involution , such that (r x)* = r′x*  ∀rR, xA.[2]

The base *-ring is usually the complex numbers (with acting as complex conjugation) and commutes with A.[clarification needed]

Since R is central in A, that is,

rx = xr   ∀rR, xA

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

(λ x + μy)* = λ′x* + μ′y*

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.,

  • f(a*) = f(a)* for all a in A.

*-operation[edit]

A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarly[vague] to taking adjoints in GLn(C).

Notation[edit]

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.

Examples[edit]

Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:

Additional structures[edit]

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

  • The Hermitian elements form a Jordan algebra;
  • The skew Hermitian elements form a Lie algebra;
  • If 2 is invertible in the *-ring, then 1/2(1 + *) and 1/2(1 − *) are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.

Skew structures[edit]

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.[vague]

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[edit]

Notes and references[edit]

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