# proof of Gelfand-Naimark representation theorem

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 ProofOfGelfandNaimarkRepresentationTheorem 2013-03-22 18:01:17 2013-03-22 18:01:17 asteroid (17536) asteroid (17536) 7 asteroid (17536) Proof msc 46L05 GelfandNaimarkRepresentationTheorem