Essential commutants and characterizations of the Toeplitz algebra

Abstract

We study the Toeplitz algebra which is generated by Toeplitz operators with bounded symbols on the Fock space Fpα. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For p=2 these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for p≠2. Instead, we use a recent result of Fulsche to generalize Xia's theorems.