Translation Invariant Pure State on $$\otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}})$$ ⊗ Z M d ( C ) and Haag Duality

Abstract

We prove Haag duality property of any translation invariant pure state on \({\mathcal B}= \otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}}), \;d \ge 2\), where \(M_d({\mathbb {C}})\) is the set of \(d \times d\) dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on \({\mathcal B}\).