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.
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- (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.
The base *-ring R is often the complex numbers (with * acting as complex conjugation).
It follows from the axioms that * on A is conjugate-linear in R, meaning
- (λ x + μ y)* = λ′ x* + μ′ y*
for λ, μ ∈ R, x, y ∈ A.
A *-homomorphism f : A → B is an algebra homomorphism that is compatible with the involutions of A and B, i.e.,
Philosophy of the *-operation
- 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 √) 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 D) 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). Note that neither 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 R is a *-algebra over R with P *(x) = P (−x).
- If (A, +, ×, *) is simultaneously a *-ring, an algebra over a ring R (commutative), and (r x)* = r (x*) ∀r ∈ R, x ∈ A, then A is a *-algebra over R (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 R, its quotient by any its *-ideal is a *-algebra over R.
- 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 K and its polynomial ring K[x]: the quotient by x = 0 restores K.
- 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).
Not every algebra admits an involution:
Regard the 2x2 matrices over the complex numbers.
Consider the following subalgebra:
Any nontrivial antiautomorphism necessarily has the form:
for any complex number .
It follows that any nontrivial antiautomorphism fails to be idempotent:
Concluding that the subalgebra admits no involution.
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/(1 + *) and 1/(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.
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.