The conjecture of causal decompositions proposes that there is an equivalence between two fundamental structures of quantum theory: causal structure and compositional structure. It would provide a dramatic connection between an empirical, operationally accessible notion (causal structure) and a powerful mathematical structure (compositional structure). This conjecture has recently been proven in the case of 1D Quantum Cellular Automata (QCAs), (which can be seen as representing lightcone-abiding dynamics in a discretised (1+1)D Minkowski spacetime), using novel and powerful mathematical techniques that employ C* algebras. The proof is however limited to a finite number of spatial sites, because the techniques have only been developed for finite-dimensional C* algebras. The project is for the students to generalise this proof to the infinite case, by investigating the extension of these techniques to the case of C* algebras of infinite dimension. This project is expected to require a significant deal of mathematics and abstraction, but the tools to be developed hold deep physical significance and promise to find application in other fields. The internship could continue into a PhD.