Subcentrality of restrictions of boundary measures on state spaces of C∗-algebras

AbstractLet F be a closed face of the weak∗ compact convex state space of a unital C∗-algebra A. The author has already shown that F is a Choquet simplex if and only if pφFπφ(A)″pφF is abelian for any φ in F with associated cyclic representation (Hφ,πφ,ξφ), where pφF is the orthogonal projection of Hφ onto the subspace spanned by vectors η defining vector states a → 〈πφ(a)η, η)〉 lying in F. It is shown here that if B is a C∗-subalgebra of A containing the unit and such that ξφ is cyclic in Hφ for πφ(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 pφF(πφ(A), πφ(B)′)″pφF is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (πφ(A)′ ∩ πφ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C∗-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central