proof of Gelfand-Naimark representation theorem

representationrepresentations

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

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

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

$\pi :𝒜⟶B\left({\displaystyle \underset{\varphi \in S(𝒜)}{\oplus }}{H}_{\varphi }\right)$

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

We now prove that this representation is injective.

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

$\varphi (a)=⟨{\pi }_{\varphi }(a){\xi }_{\varphi },{\xi }_{\varphi }⟩=0$

where ${\xi }_{\varphi }\in H_{\varphi }$ is the cyclic vector associated with ${\pi }_{\varphi }$. Since for every $\varphi \in S(𝒜)$ we have $\varphi (a)=0$, 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 $\pi (𝒜)$ is a closed *-subalgebra of $B\left({\displaystyle \underset{\varphi \in S(𝒜)}{\oplus }}{H}_{\varphi }\right)$. Thus, we have proven that $\pi$ is an isometric isomorphism between $𝒜$ and a closed *-subalgebra of $B(H)$, for a suitable Hilbert space $H$. $\mathrm{\square }$