proof of Gelfand-Naimark representation theorem
Proof: Let be a -algebra (http://planetmath.org/CAlgebra). We intend to prove that is isometrically isomorphic to a norm closed *-subalgebra of , the algebra of bounded operators of a suitable Hilbert space .
Let denote the state space of . For every state the Gelfand-Naimark-Segal construction allows one to construct a representation (http://planetmath.org/BanachAlgebraRepresentation) of in a Hilbert space .
Now consider the direct sum (http://planetmath.org/BanachAlgebraRepresentation) of these representations . Recall that is a representation
of in the direct sum of the family of Hilbert spaces (http://planetmath.org/DirectSumOfHilbertSpaces) .
We now prove that this representation is injective.
Suppose there exists such that . Then, for all , . Thus, by definition of ,
where is the cyclic vector associated with . Since for every we have , we can conclude that (see this entry (http://planetmath.org/PropertiesOfStates)), i.e. is injective.
Since an injective *-homomorphism between -algebras is isometric (http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric), we conclude that is also isometric. Hence is a closed *-subalgebra of . Thus, we have proven that is an isometric isomorphism between and a closed *-subalgebra of , for a suitable Hilbert space .
|Title||proof of Gelfand-Naimark representation theorem|
|Date of creation||2013-03-22 18:01:17|
|Last modified on||2013-03-22 18:01:17|
|Last modified by||asteroid (17536)|