*-algebra

=*-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.

*-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:$(lambda\; x+\; mu\; y)^*\; =\; lambda^*\; x^*\; +\; mu^*\; y^*$for $lambda,\; mu\; in\; R$, $x,y\; in\; A$.Proof::$(lambda\; x+\; mu\; y)^*\; =\; x^*lambda^*\; +\; y^*mu^*\; =\; lambda^*\; x^*\; +\; mu^*\; y^*$

A *-homomorphism $fcolon\; A\; o\; B$ is algebra homomorphism that is compatible with the involutions of "A" and "B", i.e.,
* $f(a^*)\; =\; f(a)^*$ for all "a" in "A".

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 $g\; mapsto\; g^\{-1\}.$

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.

kew 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.

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.

ee also

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