*-algebra
From Wikipedia, the free encyclopedia
In mathematics, a *-ring is an associative ring with an antilinear, antiautomorphism * : A → A which is an involution. More precisely, * is required to satisfy the following properties:
for all x,y in A.
A *-algebra is a *-ring that is an associative algebra over another *-ring, usually the *-ring of complex numbers (with * acting as complex conjugation).
The most obvious example of a *-algebra is the field of complex numbers C where * is just complex conjugation. 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.
An algebra homomorphism f : A → B is a *-homomorphism if it is compatible with the involutions of A and B, i.e.
- f(a * ) = f(a) * for all a in A.
An element a in A is called self-adjoint if a* = a.
