We give a classificaion of "small" monotone complete C*-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 to the power of the continuum distinct equivalence classes. This remains true when the clasification is restricted to special classes of monotone complete C*-algebras e.g. factors, injecive factors, injective operator systems and commutative algebras with separable structure space. Some examples and applications are given.
- monotone complete C*-algebras
- operator algebras