  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.
