  In this paper we study the Toeplitz algebra, which is generated by Toeplitz
operators with bounded symbols on the Fock space $F^p_{\alpha}$. 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 $p \neq 2$. Instead, we use a recent result of Fulsche to
generalize Xia's theorems.
