Type of von Neumann Algebra Associated with Free Field

Abstract

A class of von Neumann algebras associated with the normal representation (i.e. the Fock representation) of the canonical commutation relations has been studied in an earlier paper. > The von Neumann algebra in question is always the tensor product of an abelian algebra and a factor of the infinite type. A necessary and sufficient condition for the factor to be of type Icc has been obtained in reference 1). In the present paper it will be shown that the factor in question is of type IIIcc unless it is of type t'. Recently there have been some interests in the type of von Neumann algebras of local observables in quantum field theory. J,SJ,), ) The von Neumann algebras of 'local observables for a free scalar field) as well as those for a generalized free field belong to the class of von Neumann algebras considered in reference 1) and our result implies that they are either type III' or t:o. In particular, for those cases where type Icc have been excluded,J,sl, l the algebra must be of type IIICXJ . Our result also implies that the algebra of all creation and annihilation operators in quantum theory of an infinite free Bose gasl is a factor of type IIIcc when no condensation is present.