*-algebra

Template:Short description Template:Algebraic structures 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 Template:Mvar and Template:Mvar, where Template:Mvar is commutative and Template:Mvar has the structure of an associative algebra over Template:Mvar. 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.Template:Efn Template:Sister project

Definitions

*-ring

Template:Ring theory sidebar 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 Template:Mvar.

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.

*-algebra

A *-algebra Template:Mvar is a *-ring,Template:Efn with involution * that is an associative algebra over a commutative *-ring Template:Mvar with involution Template:Mvar, such that (r x)* = rTemplate:Prime x*  ∀r ∈ R, x ∈ A.

The base *-ring Template:Mvar is often the complex numbers (with Template:Mvar acting as complex conjugation).

It follows from the axioms that * on Template:Mvar is conjugate-linear in Template:Mvar, meaning

(λ x + μ y)* = λTemplate:Prime x* + μTemplate:Prime y*

for λ, μ ∈ R, x, y ∈ A.

A *-homomorphism f : A → B is an algebra homomorphism that is compatible with the involutions of Template:Mvar and Template:Mvar, i.e.,

• f(a*) = f(a)* for all Template:Mvar in Template:Mvar.

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 complex matrix algebras.

Notation

The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:

x ↦ x*, or

x ↦ x^∗ (TeX: x^*),

but not as "x∗"; see the asterisk article for details.

Examples

• Any commutative ring becomes a *-ring with the trivial (identical) involution.
• The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers C where * is just complex conjugation.
• More generally, a field extension made by adjunction of a square root (such as the imaginary unit

1. REDIRECT Template:Radic

Template:Rcat shell) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.

• A quadratic integer ring (for some Template:Mvar) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings.
• Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). None of the three is a complex algebra.
• Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation.
• The matrix algebra of n × n matrices over R with * given by the transposition.
• The matrix algebra of n × n matrices over C with * given by the conjugate transpose.
• Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
• The polynomial ring R[x] over a commutative trivially-*-ring Template:Mvar is a *-algebra over Template:Mvar with P *(x) = P (−x).
• If (A, +, ×, *) is simultaneously a *-ring, an algebra over a ring Template:Mvar (commutative), and (r x)* = r (x*)  ∀r ∈ R, x ∈ A, then Template:Mvar is a *-algebra over Template:Mvar (where * is trivial).
  • As a partial case, any *-ring is a *-algebra over integers.
• Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring.
• For a commutative *-ring Template:Mvar, its quotient by any its *-ideal is a *-algebra over Template:Mvar.
  • For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ring.
  • The same about a commutative ring Template:Mvar and its polynomial ring K[x]: the quotient by x = 0 restores Template:Mvar.
• In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial.
• The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties).

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

• The group Hopf algebra: a group ring, with involution given by g ↦ g⁻¹.

Non-Example

Not every algebra admits an involution:

Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra: $${𝒜}:=\{\left(\begin{array}{cc}a& b\\ 0& 0\end{array}\right):a,b\in \mathbb{ℂ}\}$$

Any nontrivial antiautomorphism necessarily has the form: $${\phi }_z\left[\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}1& z\\ 0& 0\end{array}\right){\phi }_z\left[\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$$ for any complex number $z\in \mathbb{ℂ}$.

It follows that any nontrivial antiautomorphism fails to be involutive: $${\phi }_z^2\left[\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\ne \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$$

Concluding that the subalgebra admits no involution.

Additional structures

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 the operators Template:Sfrac(1 + *) and Template:Sfrac(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

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 x ↦ x*.

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

• Semigroup with involution
• B*-algebra
• C*-algebra
• Dagger category
• von Neumann algebra
• Baer ring
• Operator algebra
• Conjugate (algebra)
• Cayley–Dickson construction
• Composition algebra