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The Gelfand-Naimark-Segal construction (or GNS construction) is a fundamental idea in the theory of operator algebras. It provides a procedure to construct and study representations of $C^*$ -algebras and is the first step on the proof of the Gelfand-Naimark representation theorem, which states that every $C^*$ -algebra is isometrically isomorphic to a closed *-subalgebra of $B(H)$ , the algebra of bounded operators on a Hilbert space $H$ .
There are generalizations of this construction for Banach *-algebras with an approximate unit, and some of the results stated here are in fact valid for this kind of algebras, but we will restrict our attention to the $C^*$ case.
Let $\mathcal{A}$ be a $C^*$ -algebra and $\phi$ a positive linear functional in $\mathcal{A}$ .
We are going to construct a representation $\pi_{\phi}$ of $\mathcal{A}$ and for that we need to construct a suitable Hilbert space.
Let us endow $\mathcal{A}$ with a semi-inner product defined by $\langle x, y \rangle_{\phi} : = \phi(y^*x)$ . Now we define the set
It is easily seen that $N_{\phi}$ is a closed left ideal in $\mathcal{A}$ (using the Cauchy-Schwarz inequality, which is valid in semi-inner product spaces), so that $\langle \cdot, \cdot \rangle_{\phi}$ induces a well defined inner product on the quotient $\mathcal{A}/N_{\phi}$ . The completion of $\mathcal{A}/N_{\phi}$ is then an Hilbert space, which we will be denoted by $H_{\phi}$ .
We will now define a representation of $\mathcal{A}$ on $H_{\phi}$ by left multiplication. For every $a \in \mathcal{A}$ let $\pi_{\phi}(a)$ be the operator of left multiplication by $a$ on $\mathcal{A}/N_{\phi}$ , i.e.
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Theorem 1 - The function $\pi_{\phi}(a):\mathcal{A}/N_{\phi} \longrightarrow \mathcal{A}/N_{\phi}$ defined above is linear and bounded, with $\|\pi_{\phi}(a)\| \leq \|a\|$ .
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Being bounded, the operator $\pi_{\phi}(a)$ extends uniquely to a bounded operator on $H_{\phi}$ , which we denote by the same symbol, $\pi_{\phi}(a)$ .
Let $B(H_{\phi})$ be the algebra of bounded operators on $H_{\phi}$ .
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Theorem 2 - The function $\pi_{\phi}:\mathcal{A} \longrightarrow B(H_{\phi})$ defined by $a \mapsto \pi_{\phi}(a)$ is a $C^*$ -algebra representation of $\mathcal{A}$ .
This representation is called the GNS representation associated to $\phi$ .
Suppose $\mathcal{A}$ had an identity element $e$ . In this case it is easily seen that there exists a cyclic vector $\xi_{\phi} \in H_{\phi}$ , i.e. a vector $\xi_{\phi}$ such that $\pi_{\phi}(\mathcal{A})\,\xi_{\phi}$ is dense in $H_{\phi}$ . This cyclic vector can just be chosen as $e + N_{\phi}$ .
Moreoever, this cyclic vector $\xi_{\phi} :=e + N_{\phi}$ is such that $\phi(a) = \langle \pi_{\phi}(a)\,\xi_{\phi}, \xi_{\phi}\rangle_{\phi}$ for every $a \in \mathcal{A}$ .
Thus, in this case the representation $\pi_{\phi}$ is cyclic and $\phi$ is a vector state of $\mathcal{A}$ . The result is still valid for general $C^*$ -algebras:
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Theorem 3 - Let $\pi_{\phi}$ be the representation of $\mathcal{A}$ defined previously. Then there exists a vector $\xi_{\phi} \in H_{\phi}$ such that
- $\pi_{\phi}(\mathcal{A})\,\xi_{\phi}$ is dense in $H_{\phi}$ , i.e. $\pi_{\phi}$ is cyclic,
- $\phi(a) = \langle \pi_{\phi}(a)\,\xi_{\phi}, \xi_{\phi}\rangle_{\phi}$ for every $a \in \mathcal{A}$ , i.e. $\phi$ is a vector state.
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Any pair $(\pi, \xi)$ , where $\pi$ is a representation of $\mathcal{A}$ on a Hilbert space $H$ and $\xi \in H$ , satisfying the above conditions for $\phi$ :
- $\pi(\mathcal{A})\,\xi$ is dense in $H$ ,
- $\phi(a) = \langle \pi(a)\,\xi, \xi\rangle$ for every $a \in \mathcal{A}$
is called a GNS pair for $\phi$ .
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Theorem 4 - All GNS pairs for $\phi$ are equivalent (in the sense that the corresponding representations are unitarily equivalent).
We know that are "plenty" of states on $C^*$ -algebra (see this entry), and so we have assured the existence of many (cyclic) representations. An interesting fact is that this representations associated to states are irreducible exactly when the state is a pure state:
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Theorem 5 - Let $\phi$ be a state on $\mathcal{A}$ . Then the representation $\pi_{\phi}$ is irreducible if and only if $\phi$ is a pure state.
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The fact that there are "plenty" of pure states in a $C^*$ -algebra allows one to assure the existence of irreducible representations that preserve the norm of a given element in $\mathcal{A}$ .
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Theorem 6 - Let $\mathcal{A}$ be a $C^*$ -algebra. For every element $a$ there exists an irreducible representation $\pi$ of $\mathcal{A}$ such that $\|\pi(a)\| = \|a\|$ .
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This last theorem is a fundamental step in the proof of the Gelfand-Naimark representation theorem.
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