Field algebras do not leave field domains invariant
It is proved that von Neumann algebras associated to \(Op^*\)-algebra (P,D) cannot leave the domain D of P invariant if they are type I or type III factors or finite direct sums of such factors. The class of \(Op^*\)- algebras for which this result holds includes, in particular, all (P,D) generated by essentially self-adjoint operators or such that \(P_ s'=P_ w'\) (resp. the strong and the weak bounded commutants of P); the only condition on the von Neumann algebra R is \(R\supset (P_ w')'\). The result is applied to quantum field systems described by Wightman fields A(f) with the definition domain D and von Neumann field algebras, global F and local F(0). It gives that in typical cases (in particular, for all free fields) the field algebras F and F(0) do not leave D invariant, and also that the Op\({}^*\)-algebras generated by Wightman fields cannot belong to some simplest classes of \(Op^*\)-algebras, like \(SV^*\) or closed \(EW^*\)-algebras.
\(Op^*\)- algebras generated by Wightman fields
scientific article; zbMATH DE number 3962549