*-algebra

In mathematics, and more specificially 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

*-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. Note that the third axiom is actually redundant, because the second and fourth axioms imply $1^*$ is also an identity, and identities are unique.

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

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

*-algebra

A *-algebra A is a *-ring with involution * that is an associative algebra over a commutative *-ring R with involution ', such that $(rx)^* = r'x^* \forall r \in R, x \in A$.

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

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

$(\lambda x+ \mu y)^* = \lambda^* x^* + \mu^* y^*$

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

A *-homomorphism $f\colon A \to B$ 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

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 to taking adjoints in $GL_{n}(\mathbb{C})$.

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, if a blind eye is turned on the nonassociativity of the latter.
• Another example is the matrix 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.
• 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 $g \mapsto g^{-1}.$

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.

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), neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar.

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

• B*-algebra
• C*-algebra
• von Neumann algebra
• Baer ring
• operator algebra