Essential commutants on strongly pseudo-convex domains

Consider a bounded strongly pseudo-convex domain Ω with smooth boundary in Cⁿ. Let T be the Toeplitz algebra on the Bergman space L_a²(Ω). That is, T is the C^⁎-algebra generated by the Toeplitz operators {T_f:f∈L^∞(Ω)}. Extending the work [27,28] in the special case of the unit ball, we show that on any such Ω, T and {T_f:f∈VO_bdd}+K are essential commutants of each other, where K is the collection of compact operators on L_a²(Ω). On a general Ω considered in this paper, the proofs require many new ideas and techniques. These same techniques also enable us to show that for A∈T, if 〈Ak_z,k_z〉→0 as z→∂Ω, then A is a compact operator.