## Encyclopedia > B-star-algebra

Article Content

# 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)
• 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 )

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Christiania ...   Contents Christiania Christiania can refer to: Christiania - the name of Oslo, from 1624 to 1925. The Free State of Christiania - a partially self-governing ...