Towards a Quantum Domain Theory: Order-enrichment and Fixpoints in W*-algebras

We discuss how the theory of operator algebras, also called operator theory, can be applied in quantum computer science. From a computer scientist point of view, we explain some fundamental results of operator theory and their relevance in the context of domain theory. In particular, we consider the category W∗ of W*-algebras together with normal sub-unital maps, provide an order-enrichment for this category and exhibit a class of its endofunctors with a canonical fixpoint.