Operator Product States on Tensor Powers of $C^\ast$-Algebras

The program of matrix product states on tensor powers $\mathcal A^{\otimes \mathbb Z}$ of $C^\ast$-algebras, initiated in Comm. Math. Phys. {\bf 144}, 443-490 (1992), is re-assessed in a context where $\mathcal A$ is a generic nuclear $C^\ast$-algebra. For any shift invariant state $\omega$, we demonstrate the existence of an order kernel ideal $\mathcal K_\omega$, whose quotient action reduces and factorizes the initial data $(\mathcal A^{\otimes \mathbb Z}, \omega)$ to the tuple $(\mathcal A,\mathcal B_\omega = \mathcal A^{\otimes \mathbb N^\times}/\mathcal K_\omega, \mathbb E_\omega : \mathcal A \otimes \mathcal B_\omega \to \mathcal B_\omega, \bar \omega : \mathcal B_\omega \to \mathbb C)$, where $\mathcal B_\omega$ is an operator system and $\mathbb E_\omega$ and $\bar \omega$ are unital and completely positive maps. Reciprocally, given a (input) tuple $(\mathcal A,\mathcal S,\mathbb E,\phi)$ that shares similar attributes, we supply an algorithm that produces a shift-invariant state on $\mathcal A^{\otimes \mathbb Z}$. We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite site algebras, such as the group algebra of discrete amenable groups.