B*-algebras are mathematical structures studied in
functional analysis. A B
*-algebra
A is a
Banach algebra over the field of
complex numbers, together with a map
* :
A -> A called
involution which has the follow properties:
- (x + y)* = x* + y* for all x, y in A
- (the involution of the sum of x and y is equal to the sum of the involution of x with the involution of y)
- (λ x)* = λ* x* for every λ in C and every x in A; here, λ* stands for the complex conjugation of λ.
- (xy)* = y* x* for all x, y in A
- (the involution of the product of x and y is equal to the product of the involution of x with the involution of y)
- (the involution of the involution of x is equal to x)
If the following property is also true, the algebra is actually a '''C*-algebra''':
- ||x x*|| = ||x||2 for all x in A.
- (the norm of the product of x and the involution of x is equal to the norm of x squared )