Domains of Commutative C-Subalgebras

Operator algebras provide uniform semantics for deterministic, reversible, probabilistic, and quantum computing, where intermediate results of partial computations are given by commutative sub algebras. We study this setting using domain theory, and show that a given operator algebra is scattered if and only if its associated partial order is, equivalently: continuous (a domain), algebraic, atomistic, quasi-continuous, or quasialgebraic. In that case, conversely, we prove that the Lawson topology, modelling information approximation, allows one to associate an operator algebra to the domain.