Duplicable von Neumann Algebras

Recently, we have shown that von Neumann algebras form a model for Selinger and Valiron's quantum lambda calculus. In this paper, we explain our choice of interpretation of the duplicability operator "!" by studying those von Neumann algebras that might have served as the interpretation of duplicable types, namely those that carry a (commutative) monoid structure with respect to the spatial tensor product. We show that every such monoid is the (possibly infinite) direct product of the complex numbers, and that our interpretation of the ! operator of the quantum lambda calculus is exactly the free (commutative) monoid.