An inclusion is a pair (C,D) where C is a unital C*-algebra with unit 1 and D is an abelian C*-algebra with 1 ∈ D ⊆ C. The inclusion is regular if the span of the set N_D(C):= {v ∈ C : vDv* ∪ v*Dv ⊆ D} is norm-dense in C. Regular inclusions arise naturally in many contexts, for example the image of a covariant representation of certain dynamical systems generates a regular inclusion. A C*-diagonal is a regular inclusion where every pure state of D extends uniquely to a pure state of C and the expectation E: C → D (which exists by the extension property) is faithful. C*-diagonals have a number of desirable properties, including the fact that they admit reasonable coordinate systems.

I will identify a certain ideal, rad(C, D), for an inclusion, and show that an inclusion (C, D) embeds nicely in a C*-diagonal if and only if rad(C, D)=(0).

As an application, I will show that if rad(C, D)=(0), then D norms C in the sense of Pop-Sinclair and Smith.