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
- (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
- (x*)* = x for all x in A
B* algebras are really a special case of * algebras.
If the following property is also true, the algebra is actually a C*-algebra:
- ||x x*|| = ||x||2 for all x in A.
See also: algebra, associative algebra, * algebra.