Finite-excitation vacuum spin representation and normality

Assume Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $d\ge 1$, $\mathrm{\Gamma }={\mathbb{Z}}^d$, and let the finite on-site spin dimension be $q\ge 1$, with the local matrix algebras and factor inclusions of Local spin matrix algebras and their factor inclusions and Local spin factor inclusions preserve the operator norm. Fix a reference basis label $0$ at each site. This is a relabeling of the local basis $\{1,\dots ,q\}$ used in Local spin matrix algebras and their factor inclusions; it changes no matrix algebra.

Define the finite-excitation configuration set $S_{\mathrm{f}\mathrm{i}\mathrm{n}}:=\{s:\mathrm{\Gamma }\to \{0,\dots ,q-1\}:\{x\in \mathrm{\Gamma }:s_x\mathrm{\ne }0\}\text{ is finite}\}\mathrm{.}$ This set is countable: every finite support is contained in a box $[-n,n{]}^d$, and each box has finitely many configurations that vanish outside it. The boxes form an explicit countable exhaustion, so the union of these finite configuration sets is countable. Thus its coordinate Hilbert space is separable. For $s\in S_{\mathrm{f}\mathrm{i}\mathrm{n}}$, let $e_s$ be the corresponding coordinate vector. The vacuum spin Hilbert space is ${\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}}:={\mathrm{\ell }}^2(S_{\mathrm{f}\mathrm{i}\mathrm{n}}),⟨\psi ,\phi ⟩:={\sum }_{s\in S_{\mathrm{f}\mathrm{i}\mathrm{n}}}{\psi }_s\overline{{\phi }_s},$ where the inner product is linear in its first variable. The vectors $(e_s{)}_{s\in S_{\mathrm{f}\mathrm{i}\mathrm{n}}}$ form its specified orthonormal basis, and $\mathrm{\Omega }:=e_{s^0}$, with $s_x^0=0$ for all $x$, is the reference vacuum vector.

For a finite region $\mathrm{\Lambda }$, write $S_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathrm{\Gamma }\setminus \mathrm{\Lambda }):=\{u:\mathrm{\Gamma }\setminus \mathrm{\Lambda }\to \{0,\dots ,q-1\}:\{x:u_x\mathrm{\ne }0\}\text{ is finite}\}\mathrm{.}$ Restriction to $\mathrm{\Lambda }$ and its complement gives the configuration bijection $S_{\mathrm{f}\mathrm{i}\mathrm{n}}\cong \{0,\dots ,q-1{\}}^{\mathrm{\Lambda }}\times S_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathrm{\Gamma }\setminus \mathrm{\Lambda })$ induces a unitary $U_{\mathrm{\Lambda }}:{\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}}\to {\mathcal{H}}_{\mathrm{\Lambda }}\otimes {\mathrm{\ell }}^2(S_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathrm{\Gamma }\setminus \mathrm{\Lambda }))\mathrm{.}$ For $T\in {\mathcal{A}}_{\mathrm{\Lambda }}$, define ${\pi }_{\mathrm{\Lambda }}(T):=U_{\mathrm{\Lambda }}^{-1}(T\otimes I)U_{\mathrm{\Lambda }}\mathrm{.}$ Under the direct-sum decomposition over outside configurations, ${\pi }_{\mathrm{\Lambda }}(T)$ is a copy of $T$ on each block. The outside configuration set contains the all-zero configuration, so it is nonempty; the norm of this direct sum is therefore the supremum of nonempty identical block norms, namely $\mathrm{\parallel }{\pi }_{\mathrm{\Lambda }}(T)\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$. If $\mathrm{\Lambda }\subseteq {\mathrm{\Lambda }}^{\mathrm{\prime }}$, associativity of the configuration factorization gives ${\pi }_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}(T\otimes I)={\pi }_{\mathrm{\Lambda }}(T)$ on every basis vector. Thus the local maps define a unital isometric $\ast$-representation ${\pi }_{\mathrm{l}\mathrm{o}\mathrm{c}}$ on the strictly local algebra. For a Cauchy sequence $A_n$ in that algebra, ${\pi }_{\mathrm{l}\mathrm{o}\mathrm{c}}(A_n)$ is Cauchy in operator norm; its limit is independent of the chosen representative because $\mathrm{\parallel }{\pi }_{\mathrm{l}\mathrm{o}\mathrm{c}}(A_n-A_n^{\mathrm{\prime }})\mathrm{\parallel }=\mathrm{\parallel }{A}_n-A_n^{\mathrm{\prime }}\mathrm{\parallel }$. Completeness of $\mathcal{B}({\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}})$ follows directly: an operator-norm Cauchy sequence is pointwise Cauchy on every vector, its pointwise limits define a bounded linear operator, and the same Cauchy estimates give operator-norm convergence. The limits therefore define a unique continuous extension $\pi$ to the norm completion $\mathcal{A}$. Continuity of product and adjoint and the norm identity show that the extension remains an isometric $\ast$-representation, hence is faithful. This construction selects a particular representation; it does not identify $\mathcal{A}$ with all of $\mathcal{B}({\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}})$.

For a set $\mathcal{S}\subseteq \mathcal{B}({\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}})$, its commutant is ${\mathcal{S}}^{\mathrm{\prime }}:=\{T\in \mathcal{B}({\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}}):TS=ST\text{ for every }S\in \mathcal{S}\},$ and its bicommutant is ${\mathcal{S}}^{\mathrm{\prime }\mathrm{\prime }}:=({\mathcal{S}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$. A normal state on the represented von Neumann algebra $\pi (\mathcal{A}{)}^{\mathrm{\prime }\mathrm{\prime }}$ means a positive normalized functional $\phi$ that preserves suprema of bounded increasing nets of positive operators: if $0\le X_i\uparrow X$ in $\pi (\mathcal{A}{)}^{\mathrm{\prime }\mathrm{\prime }}$, then $\phi (X)={\sup }_i\phi (X_i)$. Here $\pi (\mathcal{A}{)}^{\mathrm{\prime }\mathrm{\prime }}$ is the bicommutant in $\mathcal{B}({\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}})$; $\pi (\mathcal{A})$ is a unital $\ast$-algebra and its bicommutant is strongly closed. The term normal is representation-dependent.

The stated norm-preserving extension uses the norm completion and local inclusions, not the C*-identity of the completion. The C*-structure of the current quasi-local supplier remains conditional on its separately recorded finite-matrix input. Naaijkens constructs the spin-$1\mathrm{/}2$, one-dimensional finite-excitation representation; the finite-dimensional, ${\mathbb{Z}}^d$ definition here fixes the corresponding basis and local action explicitly.