B*-algebra

B*-algebras are mathematical structures studied in functional analysis.

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"||,

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 Because of this, the term B* algebra is rarely used in current terminology, and has been replaced by the (overloading of) the term 'C* algebra'.

ee also

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

References

*cite book | author=G. F. Simmons | title=Introduction to Topology and Modern Analysis | publisher=McGraw-Hill | year=1963 | isbn=0-07-085695-8

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