Translation invariant pure state on B=⊗j∈ZMd(j)(C) and its split property

Abstract

In this paper, we prove that a real lattice symmetric reflection positive translation-invariant pure state of B=⊗j∈ZMd(j)(C) admits split property, if and only if its two-point spatial correlation functions decay exponentially. We use amalgamated representation of Cuntz algebras to represent two-point spatial correlation functions on an augmented Hilbert space. The underling symmetries and reflection positive property of the pure state make it possible to investigate its split and decaying two-point correlation functions properties as spectral properties of a contractive self-adjoint operator on the augmented Hilbert space. Haag duality property of the pure state is crucially used in the analysis.