Some Properties of a Class of Diagonalizable States of von Neumann Algebras

In this paper, a class of representations of uniformly hyperfinite algebras is constructed and the corresponding von Neumann algebras are studied. It is proved that, under certain conditions, the Markov states generate factors of type $\operatorname{III}_\lambda$, where $\lambda\in(0,1)$, in the GNS representation; this gives a negative answer to the conjecture that the factors corresponding to Hamiltonians with nontrivial interactions have type $\operatorname{III}_1$. It is shown that, for a certain class of Hamiltonians, there exists a unique translation-invariant ground state.