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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 )

See also: algebra.



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