Theorem 2 (Gelfand–Naimark)

Let A be a stellar algebra. There exist a Hilbert space E and an isometric morphism of involutive algebras $\varphi$ of A into $\mathscr{L}(E)$. If A is unital, there exists such a unital morphism.

For every $b\in A-\{0\}$, let $\lambda_b$ be a continuous positive linear form on A such that $\lambda_b(b^*b)>0$ (Prop. 8). Since A admits an approximate unit (I, p. 121, Prop. 18), there exists a Hilbert realization $(E_b, x_b, \varphi_b)$ of $\lambda_b$ (Prop. 5). If A is unital, one may suppose that $\varphi_b$ is unital (loc. cit.). We have $\|\varphi_b(b)\|^2=\lambda_b(b^*b)\not = 0$.

Let E be the external Hilbert sum of the spaces $E_b$ for $b$ belonging to A $-\{0\}$. For every $a\in A$, let us denote by $\varphi (a)\in \mathscr{L}(E)$ the unique continuous linear mapping whose restriction to $E_b$ coincides with $\varphi_b(a)$ for every $b\in A-\{0\}$. The mapping $a\mapsto \varphi (a)$ is a morphism of involutive algebras; it is injective, hence isometric (Prop. 9 of I, p. 112), and satisfies $\varphi (1) = 1_E$ if A is unital. This completes the proof.

The category whose objects are stellar algebras and whose morphisms are morphisms of involutive algebras is therefore equivalent to the category whose objects are closed involutive subalgebras of the algebras of endomorphisms of Hilbert spaces, and whose morphisms are morphisms of involutive algebras.