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The type of von Neumann algebras associated with the quantum theory of a free Bose field is found. It is type IIIco except the case of type leo determined in an earlier paper. 1. Introduction 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. 1 > 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"'.
Huzihiro Araki (Tue,) studied this question.