2024/07/13 by Zhu, Jingming, Zhang, Chaohua
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2407.09898
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 \Tf : f∈ L∞(\mathbbD, dv)\ on Bergman space L2a(\mathbbD, dv) of the ploydisk \mathbbD in ℂn actually coincides with the Toeplitz algebra T(\mathbbD). 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(\mathbbD) also coincides with D. Based on these results, we further proved that the essential commutant of Toeplitz algebra T(\mathbbD) equals to \Tg: g∈ VObdd\ + K where VObdd is the collection of functions of vanishing oscillation on polydisk \mathbbD and K denotes the collection of compact operators on L2a(\mathbbD, dv). On the other hand, we also prove that the essential commutant of \Tg: g∈ VObdd\ is T(\mathbbD), which implies that image of T(\mathbbD) in the Calkin algebra satisfies the double commutant relation: π(T(\mathbbD))=π(T(\mathbbD))''.