B*-algebra

B*-algebras were mathematical structures studied in functional analysis. As it is now known that all B*-algebras are C*-algebras (and vice versa), the term B*-algebra is no longer widely used.

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:

1. (x + y)* = x* + y* for all x, y in A.
2. $(\lambda x)^* = \bar{\lambda}x^*$ for every λ in C and every x in A; here, $\bar{\lambda}$ denotes the complex conjugate of λ.
3. (xy)* = y* x* for all x, y in A.
4. (x*)* = x for all x in A.

In most natural examples, one also has that the involution is isometric, i.e.

• ||x*|| = ||x||,

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 Because of this, the term B* algebra is rarely used in current terminology, and has been replaced by the (overloading of) the term 'C* algebra'.

See also

• Algebra over a field
• Associative algebra
• *-algebra
• C*-algebra.

References

• G. F. Simmons (1963). Introduction to Topology and Modern Analysis. McGraw-Hill. ISBN 0-07-085695-8.