*-algebra

From Wikipedia, the free encyclopedia

1 *-ring
2 *-algebra
3 Examples
4 Additional structures
- 4.1 Skew structures
5 See also

[edit] *-ring

In mathematics, a *-ring is an associative ring with a map * : A → A which 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.

Elements such that $x * = x$ are called self-adjoint or Hermitian.

One can define a sesquilinear form over any *-ring.

[edit] *-algebra

A *-algebra A is a *-ring that is an associative algebra over another *-ring R, with the * agreeing on $R \subset A$ .

The base *-ring is usually the complex numbers (with * acting as complex conjugation).

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

(λ x + μ y) * = λ * x * + μ * y *

for $\lambda, \mu \in R$ , $x,y \in A$ . Proof:

(λ x + μ y) * = x * λ * + y * μ * = λ * x * + μ * y *

A *-homomorphism $f\colon A \to B$ is algebra homomorphism that is compatible with the involutions of A and B, i.e.,

$f (a *) = f (a) *$ for all a in A.

[edit] Examples

The most familiar example of a *-algebra is the field of complex numbers C where * is just complex conjugation.

More generally, the conjugation involution in any Cayley-Dickson algebra such as the complex numbers, quaternions and octonions.

Another example is the algebra of n×n matrices over C with * given by the conjugate transpose.

Its generalization, the Hermitian adjoint of a linear operator on a Hilbert space is also a star-algebra.

In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial.

Any commutative ring becomes a *-ring with the trivial involution.

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 $g \mapsto g^{-1}$

[edit] 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, then $\frac{1}{2}(1+*)$ and $\frac{1}{2}(1-*)$ are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. This decomposition is as a vector space, not as an algebra, because the idempotents are operators, not elements of the algebra.

[edit] Skew structures

Given a *-ring, there is also the map $x \mapsto -x^*$ . This is not a *-ring structure (unless the characteristic is 2, in which case it's identical to the original *), as $1 \mapsto -1$ (so * is not a ring homomorphism), but it satisfies the other axioms (linear, antimultiplicative, involution) and hence is quite similar.