- B*-algebra
**B*-algebras**are mathematical structures studied infunctional analysis .**General Banach *-algebras**A Banach *-algebra "A" is a

Banach algebra over the field ofcomplex number s, together with a map * : "A" → "A" called "involution" which has the following properties:

# ("x" + "y")* = "x"* + "y"* for all "x", "y" in "A".

# $(lambda\; x)^*\; =\; ar\{lambda\}x^*$ for every λ in**C**and every "x" in "A"; here, $ar\{lambda\}$ 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 Because of this, the term B* algebra is rarely used in current terminology, and has been replaced by the (overloading of) the term 'C* algebra'.**ee also***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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