Star-algebra

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:

<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>

for all x,y in A, and all z in C.

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.

An algebra homomorphism f : A → B is a *-homomorphism if it is compatible with the involutions of A and B, i.e.

<math>f(a^*) = f(a)^*<math> for all a in A.

An element a in A is called self-adjoint if a* = a.

Star-algebra

See also