Commutative C-subalgebras of Simple Stably Finite C-algebras with Real Rank Zero

Let X be a second countable, path connected, compact metric space and let A be a unital separable simple exact Z-stable real rank zero C∗-algebra. We classify all the embeddings (up to approximate unitary equivalence) of C(X) into A. Specifically, we prove the following: Theorem: Let α ∈ KL(C(X), A)+,1 and let λ : T (A) → T (C(X)) be an affine continuous map such that (i) if h ∈ Aff(T (C(X))) is such that h ≥ 0 and h is not the zero function then Aff(λ)(h)(τ) > 0 for every τ ∈ T (A); and (ii) if p is a projection in C(X)⊗K then λ(τ)(p) = τ(α(p)) for all τ ∈ T (A). Then there is a unital ∗-monomorphism φ : C(X) → A such that KL(φ) = α and T (φ) = λ. Theorem: Let φ,ψ : C(X) → A be unital ∗-monomorphisms. Then φ and ψ are approximately unitarily equivalent if and only if KL(φ) = KL(ψ) and τ ◦ φ = τ ◦ ψ for all τ ∈ T (A).