Star-algebra
From Free net encyclopedia
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. Its generalization, the Hermitian adjoint of a linear operator on a Hilbert space is also a star-algebra.
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.
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