I am considering the Hilbert space $H$ built as the infinite tensor product of separable Hilbert spaces $H_\alpha, \alpha$ in some countable or uncountable infinite set. $H$ is non separable and the operators on $H$ form a type III vN algebra.
JvN showed that $H$ breaks down into an uncountable set of separable subspaces, all orthogonal with one another - let's call them sectors. Two vectors of $H$ built as the infinite tensor products of one vector in each $H_\alpha$ are in the same sector if they differ only by a finite number of vectors in the tensor product.
I have the feeling that there is a theorem that says that bounded operators on $H$ do not connect different sectors. I have number of examples but can't find a general proof of that. Is this a well known theorem in operator algebras? (I am a physicist, not an expert in operator algebras). Or is it wrong?