B*-algebra

From Freepedia

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.