Localizations and Essential Commutant of Toeplitz Algebra on Polydisk

Usually, the norm closure of a family of operators is not equal to the C^*-algebra generated by this family of operators. But, similar with the Bergman space L²ₐ(B, dv) of the unit ball in Cⁿ, we show that the norm closure of : f∈ L∞(D, dv)\ on Bergman space L²ₐ(D, dv) of the ploydisk D in Cⁿ actually coincides with the Toeplitz algebra T(D). A key ingredient in the proof is the class of operators D recently introduced by Yi Wang and Jingbo Xia. In fact, as a by-product, we simultaneously proved that T(D) also coincides with D. Based on these results, we further proved that the essential commutant of Toeplitz algebra T(D) equals to : g∈ VObdd\ + K where VObdd is the collection of functions of vanishing oscillation on polydisk D and K denotes the collection of compact operators on L²ₐ(D, dv). On the other hand, we also prove that the essential commutant of : g∈ VObdd\ is T(D), which implies that image of T(D) in the Calkin algebra satisfies the double commutant relation: π(T(D))=π(T(D))''.