# Duplicable von Neumann Algebras

## TL;DR

This paper characterizes von Neumann algebras that can interpret the duplicability operator in quantum lambda calculus, showing they are essentially free commutative monoids formed by direct products of complex numbers.

## Contribution

It identifies the structure of von Neumann algebras suitable for modeling duplicable types in quantum lambda calculus, establishing they are free commutative monoids.

## Key findings

- Von Neumann algebras with a monoid structure are direct products of complex numbers.
- The interpretation of '!' corresponds to the free commutative monoid.
- Such monoids are characterized as possibly infinite direct products of 3.

## Abstract

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.