# Star-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:

• [itex] (x + y)^* = x^* + y^* \quad [itex]
• [itex] (z x)^* = \overline{z} x^* [itex]
• [itex] (x y)^* = y^* x^* \quad [itex]
• [itex] (x^*)^* = x \quad [itex]

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.

• [itex]f(a^*) = f(a)^*[itex] for all a in A.

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

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