Star-algebra

Encyclopedia : S : ST : STA : 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:

[ (x + y)^* = x^* + y^* \quad ]
[ (z x)^* = \overline 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. 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.

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

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

Star-algebra

See also