B*-algebra - Definition

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 following 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)

• (x^*)^* = x for all x in A

(the involution of the involution of x is equal to x)

B* algebras are really a special case of * algebras; a succinct definition is that a B*-algebra is a *-algebra that is also a Banach algebra.

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 square of the norm of x)

See also: algebra, associative algebra, * algebra.