Star-algebra (original) (raw)

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.