Unique tracial KMS state is not vacuum normal

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$, the local spin algebra, and the zero onsite model with the energy, time, positive hbar and inverse-energy beta conventions just specified.

Countable Choice is the completion assumption carried by Local spin matrix algebras and their factor inclusions and Unique onsite product Gibbs KMS state (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Unique onsite product Gibbs KMS state: under ${\mathrm{A}\mathrm{C}}_{\omega }$ and its finite-matrix hypotheses [A2], for $h=0$, $r=q^{-1}I$ and the unique KMS state has local density $q^{-\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{I}_{\mathrm{\Lambda }}$. Those finite-matrix hypotheses are the clauses in [F3] below.

Thermal product state has no normal vacuum extension: under ${\mathrm{A}\mathrm{C}}_{\omega }$ and the same finite-matrix hypotheses retained from [F1], if $q\ge 2$ and the onsite Gibbs density is full rank, this KMS state has no normal extension to $\mathcal{B}({\mathcal{H}}_{\mathrm{v}\mathrm{a}\mathrm{c}})$.

Local spin matrix algebras and their factor inclusions, Definition: every finite region $\mathrm{\Lambda }$ has a nonzero complex Hilbert space ${\mathcal{H}}_{\mathrm{\Lambda }}$ of dimension $q^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}$, with inner product linear in the first argument, and ${\mathcal{A}}_{\mathrm{\Lambda }}=\mathcal{B}({\mathcal{H}}_{\mathrm{\Lambda }})$ with the induced operator norm. Its finite-matrix clauses give $\mathrm{\parallel }{T}^{\ast }T\mathrm{\parallel }=\mathrm{\parallel }T{\mathrm{\parallel }}^2$; positive complex-linear functionals are exactly $T\mapsto {\mathrm{Tr}}_{\mathrm{\Lambda }}(\rho T)$ for unique $\rho \ge 0$, with functional norm ${\mathrm{Tr}}_{\mathrm{\Lambda }}\rho$, so normalization is equivalent to trace one. These clauses are supplied by the proved finite-matrix lemma cited in that Definition, including the empty region of dimension one. Every pair of local observables embeds into a common finite-region matrix algebra, where the finite trace is cyclic.

Norm-completed quasi-local quantum spin algebra: the strictly local algebra is ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}={\bigcup }_{\mathrm{\Lambda }\subseteq \mathrm{\Gamma },\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }<\mathrm{\infty }}{\mathcal{A}}_{\mathrm{\Lambda }}$, and the quasi-local algebra is its norm completion $\mathcal{A}={\overline{{\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}}}^{\mathrm{\parallel }\cdot \mathrm{\parallel }}$; therefore strictly local observables are norm dense in $\mathcal{A}$.