Sequential compactness of the quasi-local spin state space

For each $m\in \mathbb{N}$, let $R_m$ be the set of ranges of all $m$-tuples in ${\mathrm{\Gamma }}^m$, including tuples with repeated entries. Since ${\mathrm{\Gamma }}^m$ is at most countable by [F1], a surjection $\mathbb{N}\to {\mathrm{\Gamma }}^m$ composed with the range map shows $R_m$ is at most countable by [F2]. Every finite region is the range of a tuple of its cardinality, with the empty region the range of the unique $0$-tuple. Thus all finite regions form the countable union ${\bigcup }_m{R}_m$, which is at most countable by [F1] and [A1]. For each region $\mathrm{\Lambda }$, rational real and imaginary matrix entries give a countable subset of ${\mathcal{A}}_{\mathrm{\Lambda }}$: it is a finite Cartesian power of $\mathbb{Q}$, so [F1]–[F2] apply. These matrices are norm dense: for a $D_{\mathrm{\Lambda }}\times D_{\mathrm{\Lambda }}$ error matrix $E$ whose entries have modulus at most $\delta$, Cauchy–Schwarz gives $\mathrm{\parallel }Ev{\mathrm{\parallel }}_2^2\le {\sum }_{i,j}\mathrm{\mid }{E}_{ij}{\mathrm{\mid }}^2\mathrm{\parallel }v{\mathrm{\parallel }}_2^2\le D_{\mathrm{\Lambda }}^2{\delta }^2\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$, so $\mathrm{\parallel }E{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le D_{\mathrm{\Lambda }}\delta$. A second countable union, using [A1], therefore gives a countable norm-dense set $D$ in ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$, hence in its completion $\mathcal{A}$.

By [F2], choose a surjection $s:\mathbb{N}\to D$ and enumerate its values as $D_1,D_2,\dots$. For each fixed $j\ge 1$, the scalar sequence $z_n^{(j)}:={\omega }_n(D_j)$ is bounded: $D_j$ is local, and the finite-dimensional state/density interface [F3] gives $\mathrm{\mid }{\omega }_n(D_j)\mathrm{\mid }\le \mathrm{\parallel }{D}_j\mathrm{\parallel }$. The same estimate extends each ${\omega }_n$ from the dense local algebra to all of $\mathcal{A}$, so $\mathrm{\parallel }{\omega }_n\mathrm{\parallel }=1$ by normalization.

We specify a deterministic convergent-subsequence rule for any bounded complex sequence $z_n$. Choose $M>0$ with $z_n\in [-M,M{]}^2$ under $\mathbb{C}\cong {\mathbb{R}}^2$. Subdivide the current closed square into its four closed equal subsquares in fixed lexicographic order; at least one contains infinitely many terms, and the rule takes the first such square. At the first stage choose the least index whose term lies in that square; at every later stage choose the least index exceeding the previous one whose term lies in the selected square. Writing the nested squares as $I_j\times J_j$, their closed coordinate intervals are nested and have lengths tending to zero. [F5] gives unique $x\in {\bigcap }_j{I}_j$ and $y\in {\bigcap }_j{J}_j$, so $(x,y)$ is the unique common point of the squares. The subsequence terms after stage $j$ lie in the $j$th square, whose diameter tends to zero, hence converge to $(x,y)$. Every choice in this rule is determined by a finite ordering or a least natural index; it uses no additional choice axiom.

Apply the deterministic rule of step 1.3 recursively to the coordinate sequences $z_n^{(j)}$ along nested subsequences for $j\ge 1$. More precisely, start with $i_0(n)=n$; at stage $j$, write the resulting subsequence as $i_j(n)=i_{j-1}(s_j(n))$ with $s_j$ strictly increasing, so $s_j(j)\ge j$. By [F5] and recursion, these maps are defined for every positive $j$. Put $r_j=i_j(j)$ for $j\ge 1$. For $j\ge 2$, $r_j=i_{j-1}(s_j(j))\ge i_{j-1}(j)>i_{j-1}(j-1)=r_{j-1}$, so $(r_j{)}_{j\ge 1}$ is increasing. For fixed $m\ge 1$ and $j\ge m$, nestedness gives $r_j=i_m(t_{m,j})$ where $t_{m,j}\ge j$; hence these are tails of the convergent coordinate subsequence selected at stage $m$, and ${\omega }_{r_j}(D_m)$ converges.

For any $A\in {\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ and $\epsilon >0$, choose $D_m$ with $\mathrm{\parallel }A-D_m\mathrm{\parallel }<\epsilon \mathrm{/}4$ by density. Since each ${\omega }_{r_j}$ has norm one, $\mathrm{\mid }{\omega }_{r_j}(A)-{\omega }_{r_k}(A)\mathrm{\mid }\le 2\mathrm{\parallel }A-D_m\mathrm{\parallel }+\mathrm{\mid }{\omega }_{r_j}(D_m)-{\omega }_{r_k}(D_m)\mathrm{\mid }\mathrm{.}$ The last term tends to zero as $j,k\to \mathrm{\infty }$ by step 2.1. Thus ${\omega }_{r_j}(A)$ is Cauchy and converges for every local $A$. Define ${\omega }_0(A):={\lim }_j{\omega }_{r_j}(A)$ on ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$. For local $A,B$ and $\lambda \in \mathbb{C}$, the equalities ${\omega }_{r_j}(A+\lambda B)={\omega }_{r_j}(A)+\lambda {\omega }_{r_j}(B)$ pass to the limit, and ${\omega }_0(I)=1$. For every local $X$, ${\omega }_0(X^{\ast }X)={\lim }_j{\omega }_{r_j}(X^{\ast }X)\ge 0$. Finally $\mathrm{\mid }{\omega }_0(A)\mathrm{\mid }={\lim }_j\mathrm{\mid }{\omega }_{r_j}(A)\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }$. Thus ${\omega }_0$ is a bounded normalized positive functional on ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$.

On each finite region $\mathrm{\Lambda }$, write ${\rho }_{\mathrm{\Lambda }}^{(j)}$ for the density of ${\omega }_{r_j}{\mathrm{\mid }}_{{\mathcal{A}}_{\mathrm{\Lambda }}}$. For each matrix unit $E_{ab}$, its entries are rational, so $E_{ba}\in D$; the trace-pairing formula gives $({\rho }_{\mathrm{\Lambda }}^{(j)}{)}_{ab}={\omega }_{r_j}(E_{ba})$. Thus every matrix coefficient converges by step 2.1. The entrywise limit ${\rho }_{\mathrm{\Lambda }}$ is Hermitian and trace one because these are finite linear equalities; for every vector $v$, $⟨{\rho }_{\mathrm{\Lambda }}v,v⟩={\lim }_j⟨{\rho }_{\mathrm{\Lambda }}^{(j)}v,v⟩\ge 0$, so it is positive. Every partial-trace consistency equation is a finite linear equation in these entries and therefore passes to the limit. Hence $\{{\rho }_{\mathrm{\Lambda }}\}$ is a consistent family.

By [F6], the consistent family of step 4.1 extends to a unique state $\omega$ on $\mathcal{A}$, and $\omega {\mathrm{\mid }}_{{\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}}={\omega }_0$. For any $A\in \mathcal{A}$ and $\epsilon >0$, choose local $B$ with $\mathrm{\parallel }A-B\mathrm{\parallel }<\epsilon \mathrm{/}4$. Then $\mathrm{\mid }{\omega }_{r_j}(A)-\omega (A)\mathrm{\mid }\le 2\mathrm{\parallel }A-B\mathrm{\parallel }+\mathrm{\mid }{\omega }_{r_j}(B)-\omega (B)\mathrm{\mid },$ whose right side is eventually less than $\epsilon$ by step 3.1. Therefore ${\omega }_{r_j}(A)\to \omega (A)$ for every $A\in \mathcal{A}$, proving sequential compactness in the pointwise topology.