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)
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.
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