Joint mean oscillation and local ideals in the toeplitz algebra II: Local commutivity and essential commutant

A well-known theorem of Sarason asserts that if [T_f, T_h] is compact for every h ∈ H^∞, then f ∈ H^∞ + C(T). Using local analysis in the full Toeplitz algebra J = J(L^∞), we show that the membership f ∈ H^∞ + C(T) can be inferred from the compactness of a much smaller collection of commutators [T_f, T_h]. Using this strengthened result and a theorem of Davidson, we construct a proper C*-subalgebra J(ℒ) of J which has the same essential commutant as that of J. Thus the image of J(ℒ) in the Calkin algebra does not satisfy the double commutant relation. We will also show that no separable subalgebra S of J is capable of conferring the membership f ∈ H^∞ + C(T) through the compactness of the commutators {[T_f, S]: S ∈ S}.