Operator product states on tensor powers of \(C^*\)-algebras

Abstract

The program of matrix product states on tensor powers \({\mathcal {A}}^{\otimes {\mathbb {Z}}}\) of \(C^*\)-algebras is carried under the assumption that \({\mathcal {A}}\) is an arbitrary nuclear C*-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 \(({\mathcal {A}},{\mathcal {B}}_\omega = {\mathcal {A}}^{\otimes {\mathbb {N}}^\times }/{\mathcal {K}}_\omega , {\mathbb {E}}_\omega : \text{\AA }\otimes {\mathcal {B}}_\omega \rightarrow {\mathcal {B}}_\omega , {\bar{\omega }}: {\mathcal {B}}_\omega \rightarrow {\mathbb {C}})\), 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 dimensional site algebras \({\mathcal {A}}\), such as the \(C^*\)-algebras of discrete amenable groups.