In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear antiautomorphism *:A->A which is an involution. More precisely, * is required to satisfy the following properties: for all a,b in A,
- ( a + b )* = a* + b*,
- (z a)* = z* a* if z is any complex number, and
- (ab)* = b* a*
- a**=a
The field of complex numbers C is a *-algebra with * being complex conjugation.
An algebra homomorphism f:A->B is a *-homomorphism if, in addition, is compatible with the involutions of A and B. What this means is that
- f(a*)=f(a)* for all a in A.
If a*=a, then a is called self-adjoint.
See also B* algebra, C* algebra, operator algebra, self-adjoint.
