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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, and z a complex number,
- <math> (x + y)^* = x^* + y^* \quad <math>
- <math> (z x)^* = \overline{z} x^* <math>
- <math> (x y)^* = y^* x^* \quad <math>
- <math> (x^*)^* = x \quad <math>
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
- <math> f(a^*) = f(a)^*<math> for all a in A.
If a*=a, then a is called self-adjoint.
See also B* algebra, C* algebra, operator algebra, self-adjoint.
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