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

The program of matrix product states on the infinite tensor product ${mathcal A}^{otimes mathbb Z}$, initiated by Fannes, Nachtergaele and Werner in their seminal paper Commun. Math. Phys. Vol. 144, 443-490 (1992), is re-assessed in a context where $mathcal A$ is an infinite nuclear $C^ast$-algebra. While this setting presents new technical challenges, fine advances on ordered spaces by Kavruk, Paulsen, Todorov and Tomforde enabled us to push through most of the program and to demonstrate that the matrix product states accept generalizations as operator product states.