# B-star-algebra

**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)

- (
*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.

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 )

See also: algebra, associative algebra, * algebra.