  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.
