B*-algebra
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B*-algebras are mathematical structures studied in functional analysis.
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[edit] General Banach *-algebras
A Banach *-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.
for every λ in C and every x in A; here, 
denotes the complex conjugate of λ.- (xy)* = y* x* for all x, y in A.
- (x*)* = x for all x in A.
In most natural examples, one also has that the involution is isometric, i.e.
- ||x*|| = ||x||,
[edit] B* algebras
A B*-algebra is a Banach *-algebra in which the involution satisfies the following further property:
- ||x x*|| = ||x||2 for all x in A.
By a theorem of Gelfand and Naimark, given a B* algebra A there exists a Hilbert space H and an isometric *-homomorphism from A into the algebra B(H) of all bounded linear operators on H. Thus every B* algebra is isometrically *-isomorphic to a C*-algebra. Because of this, the term B* algebra is rarely used in current terminology, and has been replaced by the (overloading of) the term 'C* algebra'.
[edit] See also
[edit] References
- G. F. Simmons (1963). Introduction to Topology and Modern Analysis. McGraw-Hill. ISBN 0-07-085695-8.
