Proposition: Coherent central sectors

Suppose $e_{\alpha ,n}$ are a fixed finite family of central projections of ${\mathfrak{A}}_n(O)$, pairwise orthogonal and summing to one. Then $[e_{\alpha ,n}]$ are central projections of ${\mathfrak{A}}_{\infty }(O)$. If ${\omega }_n(e_{\alpha ,n})\to p_{\alpha }$, the limit snapshot retains the weights $p_{\alpha }$. A sector projection is nonzero in the abstract quotient whenever it is nonzero at infinitely many scales.

Proof. All projection, orthogonality, completeness, and commutation identities pass coordinatewise. The norm of a nonzero projection is one, so (eq:quotientnorm) gives the last assertion. Proposition prop:limitstates gives the weights.