Gaps in the poset of projections in the Calkin algebra

We study the gap structure of the partial order of projections of the Calkin algebra of a complex, separable, infinite-dimensional Hilbert space. We prove the existence of an analytic Hausdorff gap in this partial order. As a consequence we obtain that under Todorcevic’s Axiom and MA the gap spectrum of P(C(H)) is strictly bigger than the gap spectrum of P(ω)/Fin.