proof of Gelfand-Naimark representation theorem

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Proof: Let $\mathcal{A}$ be a $C^{*}$ -algebra (http://planetmath.org/CAlgebra). We intend to prove that $\mathcal{A}$ is isometrically isomorphic to a norm closed *-subalgebra of $B(H)$ , the algebra of bounded operators of a suitable Hilbert space $H$ .

Let $S(\mathcal{A})$ denote the state space of $\mathcal{A}$ . For every state $\phi\in S(\mathcal{A})$ the Gelfand-Naimark-Segal construction allows one to construct a representation (http://planetmath.org/BanachAlgebraRepresentation) $\pi_{\phi}:\mathcal{A}\longrightarrow B(H_{\phi})$ of $\mathcal{A}$ in a Hilbert space $H_{\phi}$ .

Now consider the direct sum (http://planetmath.org/BanachAlgebraRepresentation) of these representations $\displaystyle\pi:=\bigoplus_{\phi\in S(\mathcal{A})}\pi_{\phi}$ . Recall that $\pi$ is a representation

\displaystyle\pi:\mathcal{A}\longrightarrow B\Big{(}\bigoplus_{\phi\in S(% \mathcal{A})}H_{\phi}\Big{)}

of $\mathcal{A}$ in the direct sum of the family of Hilbert spaces (http://planetmath.org/DirectSumOfHilbertSpaces) $\{H_{\phi}\}_{\phi\in S(\mathcal{A})}$ .

We now prove that this representation is injective.

Suppose there exists $a\in\mathcal{A}$ such that $\pi(a)=0$ . Then, for all $\phi\in S(\mathcal{A})$ , $\pi_{\phi}(a)=0$ . Thus, by definition of $\pi_{\phi}$ ,

\displaystyle\phi(a)=\langle\pi_{\phi}(a)\xi_{\phi},\xi_{\phi}\rangle=0

where $\xi_{\phi}\in H_{\phi}$ is the cyclic vector associated with $\pi_{\phi}$ . Since for every $\phi\in S(\mathcal{A})$ we have $\phi(a)=0$ , we can conclude that $a=0$ (see this entry (http://planetmath.org/PropertiesOfStates)), i.e. $\pi$ is injective.

Since an injective *-homomorphism between $C^{*}$ -algebras is isometric (http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric), we conclude that $\pi$ is also isometric. Hence $\pi(\mathcal{A})$ is a closed *-subalgebra of $\displaystyle B\Big{(}\bigoplus_{\phi\in S(\mathcal{A})}H_{\phi}\Big{)}$ . Thus, we have proven that $\pi$ is an isometric isomorphism between $\mathcal{A}$ and a closed *-subalgebra of $B(H)$ , for a suitable Hilbert space $H$ . $\square$

Title	proof of Gelfand-Naimark representation theorem
Canonical name	ProofOfGelfandNaimarkRepresentationTheorem
Date of creation	2013-03-22 18:01:17
Last modified on	2013-03-22 18:01:17
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	7
Author	asteroid (17536)
Entry type	Proof
Classification	msc 46L05
Related topic	GelfandNaimarkRepresentationTheorem