B*-algebras are mathematical structures studied in functional analysis. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For functional analysis as used in psychology see the Functional analysis (psychology article 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. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
- (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.
- (x*)* = x for all x in A.
- ||x*|| = ||x||, i. e. , the involution is compatible with the norm.
B* algebras are really a special case of * algebras; a succinct definition is that a B*-algebra is a Banach *-algebra for which (5) also holds. -ring In Mathematics, a *-ring is an Associative ring with a map *: A &rarr A which is an Antiautomorphism
If the following property is also true, the algebra is actually a C*-algebra:
- ||x x*|| = ||x||2 for all x in A. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics.
See also
- Algebra over a field
- Associative algebra
- *-algebra. In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive -ring In Mathematics, a *-ring is an Associative ring with a map *: A &rarr A which is an Antiautomorphism
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