B-star-algebra
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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.