Transfer-generated infinite-chain state and correlations

The finite coefficient sum defining ${\mathcal{E}}_O(X)$ is complex-linear in both $O$ and $X$. For $O\ge 0$, write its spectral decomposition as $O={\sum }_{\alpha }\mathrm{\mid }{v}_{\alpha }⟩⟨v_{\alpha }\mathrm{\mid }$, absorbing the square roots of the nonnegative eigenvalues into $v_{\alpha }$, and put $v_{\alpha ,I}=⟨v_{\alpha },e_I⟩$. Then $O_{I,J}={\sum }_{\alpha }{v}_{\alpha ,I}\overline{v_{\alpha ,J}}$, so with $K_{\alpha }:={\sum }_J\overline{v_{\alpha ,J}}{A}^J$ the insertion map satisfies ${\mathcal{E}}_O(X)={\sum }_{\alpha }{K}_{\alpha }X{K}_{\alpha }^{\ast }\ge 0$ whenever $X\ge 0$. Diagonalizing $\mathrm{\ell }$ by [F7] and expanding $\mathrm{tr}(\mathrm{\ell }{\mathcal{E}}_O(\rho ))$ as a weighted sum of nonnegative diagonal expectations shows ${\omega }_k(O)\ge 0$. For $O=I_{d^k}$, $O_{I,J}=1$ if $I=J$ and $0$ otherwise, so the insertion sums over all length-$k$ words and equals ${\mathcal{E}}^k$. The fixed-point equations and $\mathrm{tr}(\mathrm{\ell }\rho )=1$ give ${\omega }_k(I)=\mathrm{tr}(\mathrm{\ell }{\mathcal{E}}^k(\rho ))=1$. Thus each contiguous-block functional is positive and normalized.

Tensor-factor order in the definition gives ${\mathcal{E}}_{O\otimes P}={\mathcal{E}}_O\circ {\mathcal{E}}_P$ for consecutive blocks: expanding both sides gives the same ordered product $A^{J_O}{A}^{J_P}X(A^{I_P}{)}^{\ast }(A^{I_O}{)}^{\ast }$. Inserting an identity site on the right composes with $\mathcal{E}$ on the input, which leaves $\rho$ fixed; inserting one on the left composes with $\mathcal{E}$ on the output, which leaves the trace pairing with $\mathrm{\ell }$ fixed. Repeating these identities shows that the value is independent of the chosen containing interval and that restrictions to nested finite regions agree. A positive observable remains positive when identity factors are inserted, so the local functionals on arbitrary finite regions are positive and normalized as well. The empty-region convention is consistent because every local functional assigns one to the identity.

By [F2]–[F4] each local positive normalized functional has a unique trace-one density ${\rho }_{\mathrm{\Lambda }}$. For nested regions, step 2.1 gives ${\mathrm{tr}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}((O\otimes I){\rho }_{{\mathrm{\Lambda }}^{\mathrm{\prime }}})={\mathrm{tr}}_{\mathrm{\Lambda }}(O{\rho }_{\mathrm{\Lambda }})$ for every $O\in {\mathcal{A}}_{\mathrm{\Lambda }}$. Applying the defining partial-trace pairing [F5] shows ${\mathrm{tr}}_{\mathrm{\Lambda }}(O{\mathrm{Tr}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}\setminus \mathrm{\Lambda }}({\rho }_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}))={\mathrm{tr}}_{\mathrm{\Lambda }}(O{\rho }_{\mathrm{\Lambda }})$ for all $O$. Taking $O$ to be each matrix unit makes every matrix entry of the two matrices equal, so the density family is consistent.

Apply [F6] to the consistent finite-region density family from step 3.1. Its hypotheses hold by [A1], [F2], [F3], and the positive trace-one property just proved. It yields a unique infinite algebraic state on the specified quasi-local algebra, with exactly these local restrictions and no chosen global Hilbert-space representation. This establishes that the transfer contractions in the Definition are well defined.