Subcentrality of restrictions of boundary measures on state spaces of C1-algebras

Let F be a closed face of the weak¹ compact convex state space of a unital C¹-algebra A. The author has already shown that F is a Choquet simplex if and only if p_φ^Fπ_φ(A)″p_φ^F is abelian for any φ in F with associated cyclic representation ($\text{H}$_φ,π_φ,ξ_φ), where p_φ^F is the orthogonal projection of $\text{H}$_φ onto the subspace spanned by vectors η defining vector states a → 〈π_φ(a)η, η)〉 lying in F. It is shown here that if B is a C¹-subalgebra of A containing the unit and such that ξ_φ is cyclic in $\text{H}$_φ for π_φ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if p_φ^F(π_φ(A), π_φ(B)′)″p_φ^F is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (π_φ(A)′ ∩ π_φ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C¹-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central.