On aC*-algebra approach to phase transition in the two-dimensional Ising model

We investigate the state on theC*-algebra of Pauli spins on a one-dimensional lattice (infinitely extended in both directions) which gives rise to the thermodynamic limit of the Gibbs ensemble in the two-dimensional Ising model (with nearest neighbour interaction). It is shown that the representation of the Pauli spin algebra associated with the state is factorial above and at the known critical temperature, while it has a two-dimensional center below the critical temperature. As a technical tool, we derive a general criterion for a state of the Pauli spin algebra corresponding to a Fock state of the Fermion algebra to be primary. We also show that restrictions of two quasifree states of the Fermion algebra to its even part are equivalent if and only if the projection operatorsE ₁ andE ₂ (on the direct sum of two copies of the basic Hilbert space) satisfy the following two conditions: (1)E ₁ ?E ₂ is in the Hilbert-Schmidt class, (2)E ₁ ∧ (1 ?E ₂) has an even dimension, where the even-oddness of dimE ₁ ∧ (1 ?E ₂) is called ?₂-index ofE ₁ andE ₂ and is continuous inE ₁ andE ₂ relative to the norm topology.