arXiv subjects
Publications and source records attributed to Abraham A. Westerbaan.
In this dissertation we study the category of completely positive normal contractive maps between von Neumann algebras. It includes an extensive introduction to the basic theory of $C^*$-algebras and von Neumann algebras.
Measure and integral are two closely related, but distinct objects of study. Nonetheless, they are both real-valued lattice valuations: order preserving real-valued functions $ϕ$ on a lattice $L$ which are modular, i.e., $ϕ(x)+ϕ(y) = ϕ(x\wedge y)+ϕ(x\vee y)$ for all $x,y \in L$. We unify measure and integral by developing a theory for lattice valuations. We allow these lattice valuations to take their values from the reals, or any suitable ordered Abelian group.
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.