Finitely Correlated Pure States

We study a w*-dense subset of the translation invariant states on an infinite tensor product algebra ⊗ZA, where A is a matrix algebra. These "finitely correlated states" are explicitly constructed in terms of a finite dimensional auxiliary algebra B and a completely positive map E: A⊗B → B. We show that such a state ω is pure if and only if it is extremal periodic and its entropy density vanishes. In this case the auxiliary objects B and E are uniquely determined by ω, and can be expressed in terms of an isometry between suitable tensor product Hilbert spaces.