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# Star-algebra

(Redirected from *-algebra)

In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear, antiautomorphism * : AA which is an involution. More precisely, * is required to satisfy the following properties:

• $(x + y)^* = x^* + y^* \quad$
• $(z x)^* = \overline{z} x^*$
• $(x y)^* = y^* x^* \quad$
• $(x^*)^* = x \quad$

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 : AB 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.