CAR over a Hilbert Space Has the Universal Inductive-Limit Norm

For each finite-dimensional $F\subseteq H$, the restriction $C{\mathrm{\mid }}_F$ satisfies the defining relations of ${\mathsf{C}\mathsf{A}\mathsf{R}}^{\mathrm{a}\mathrm{l}\mathrm{g}}(F)$. Its universal property gives a unital $\ast$-homomorphism ${\rho }_F$ from that algebra to $\mathcal{B}(K)$. These homomorphisms agree on inclusions because each sends every $c(f)$ to $C(f)$. By [F2], every ${\rho }_F$ is isometric since $K\mathrm{\ne }\{0\}$. They therefore define an isometric $\ast$-homomorphism ${\rho }_0$ on the algebraic union.

Since $K$ is Hilbert, it is Banach; applying [F5] with $X=Y=K$ shows $\mathcal{B}(K)$ is complete for its operator norm [F6]. The isometry ${\rho }_0$ sends Cauchy sequences to Cauchy sequences, so for $x\in \mathsf{C}\mathsf{A}\mathsf{R}(H)$ choose a local sequence $x_m\to x$ and define $\rho (x)={\lim }_m{\rho }_0(x_m)$. Isometry makes this limit independent of the chosen sequence and proves that $\rho$ is a linear isometry extending ${\rho }_0$.

Let $x,y\in \mathsf{C}\mathsf{A}\mathsf{R}(H)$ and choose local sequences $x_m\to x$ and $y_m\to y$. Since $(x_m)$ is bounded, [F9] gives $\mathrm{\parallel }{x}_m{y}_m-xy\mathrm{\parallel }\le \mathrm{\parallel }{x}_m\mathrm{\parallel }\mathrm{\parallel }{y}_m-y\mathrm{\parallel }+\mathrm{\parallel }{x}_m-x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }\to 0$; [F8] gives the same estimate for ${\rho }_0(x_m){\rho }_0(y_m)\to \rho (x)\rho (y)$. Because ${\rho }_0(x_m{y}_m)={\rho }_0(x_m){\rho }_0(y_m)$, limits prove multiplicativity. Also [F9] gives $x_m^{\ast }\to x^{\ast }$, while [F7] and ${\rho }_0(x_m)\to \rho (x)$ give ${\rho }_0(x_m{)}^{\ast }\to \rho (x{)}^{\ast }$. Since ${\rho }_0(x_m^{\ast })={\rho }_0(x_m{)}^{\ast }$, taking limits proves $\rho (x^{\ast })=\rho (x{)}^{\ast }$. The common unit is preserved on the dense union. Any other continuous extension agrees there and hence equals $\rho$.

By [F3] the Fock space is nonzero, and under [A1] the creation and adjoint operators in [F4] give one of the stated CAR families. Applying the extension construction of steps 1.1, 2.1 and 3.1 to this family gives its representation of $\mathsf{C}\mathsf{A}\mathsf{R}(H)$. Every such representation restricts on a finite stage to a unital representation of that matrix algebra, so [F2] makes its norm on each local element equal the stage norm. Thus the global family is nonempty and the supremum of represented norms equals the inductive-limit norm for every local element.