Continuous Dcpos in Quantum Computing

We show that the unit interval of a directed-complete C*-algebra is a continuous dcpo only if the C*-algebra is a product of finite-dimensional matrix algebras. Since Selinger showed that the positive cone of any product of finite-dimensional
matrix algebras is continuous as a bounded directed-complete poset (in the process of showing that the categories CPM and Q are enriched in continuous dcpos) this is actually if and only if, and therefore shows that Selinger’s result is the best possible.
This means that attempts to define a quantum domain theory using infinite-dimensional W*-algebras or von Neumann algebras necessarily involve dcpos that are not continuous.
We also show that even when the positive cone is continuous, the Scott and Lawson topologies are not suitable for computational realizability in the noncommutative case because the positive cone of a non-commutative matrix algebra does not have a countable base.