OAI.BoundedRecovery.bounded_recovery

The theorem states that, for a standard modular datum S on a complex Hilbert space H (a von Neumann algebra M with a unit cyclic and separating vector ξ, a nondegenerate real spectral calculus D with unitary group D.unitary t, an antilinear isometric involution J, and a closed Tomita graph of a ↦ (aξ, a*ξ) over a ∈ M equal to the pairs (p, Jq) where p is mapped to q by the half-exponential graph of D) satisfying ScalarCentralizer (every a in M fixed by conjugation with D.unitary t for all real t is a complex multiple of the identity), the following holds. Let ω be an ultrafilter on ℕ that refines the at-infinity filter (ω ≤ atTop), let T be a bounded complex-linear operator on H, let s be real, and let δₙ > 0 tend to 0. Suppose hₙ are unit vectors, each with spectral support in the interval [s − δₙ, s + δₙ] (every bounded continuous function vanishing on that interval annihilates hₙ under the calculus), and suppose the limsup over n of the ω-limit of the symmetric time averages (1/2k)∫{−k}^{k} ‖T(D.unitary t hₙ)‖² dt is strictly positive. Then the recovery conclusion holds: there are a strictly increasing sequence nⱼ, operators vⱼ in M, and constants C and η > 0 such that for every j, ‖vⱼ‖ ≤ C, ‖T(vⱼ ξ)‖ ≥ η, and vⱼξ has spectral support in [s − 4δ{nⱼ}, s + 4δ_{nⱼ}]. The proof is admitted, not supplied.