On the structure of the von Neumann algebras generated by local functions of the free bose field

It is shown that the von Neumann algebra(R_mathfrak{B} )(B) generated by any scalar local functionB(x) of the free fieldA ₀(x) is equal either to(R_mathfrak{B} )(A ₀) or to(R_mathfrak{B} )(:A ₀ ² :). The latter statement holds if the state space space(mathfrak{H}_B ) obtained from the vacuum state by repeated application ofB(x) is orthogonal to the one particle subspace. In the proof of these statements, space-time limiting techniques are used.