A Lattice of Von Neumann Algebras Associated with the Quantum Theory of a Free Bose Field

Abstract

Von Neumann algebras associated with the normal representation of canonical commutation relations are studied. Corresponding to each subspace of a real Hilbert space (test function space), a von Neumann algebra on another complex Hilbert space (the Fock space) is defined. This correspondence is proved to be an isomorphism between a certain complemented lattice of subspaces and that of the von Neumann algebras. This result has an application to the duality theorem in the theory of a free scalar field, which is to be discussed in a separate paper. A necessary and sufficient condition on a subspace, in order that the corresponding von Neumann algebra is of type I, is obtained.