Localization and the Toeplitz Algebra on the Bergman Space

Let Tf denote the Toeplitz operator with symbol function f on the Bergman space La(B, dv) of the unit ball in C . It is a natural problem in the theory of Toeplitz operators to determine the norm closure of the set {Tf : f ∈ L∞(B, dv)} in B(La(B, dv)). We show that the norm closure of {Tf : f ∈ L∞(B, dv)} actually coincides with the Toeplitz algebra T , i.e., the C∗-algebra generated by {Tf : f ∈ L∞(B, dv)}. A key ingredient in the proof is the class of weakly localized operators recently introduced by Isralowitz, Mitkovski and Wick. Our approach simultaneously gives us the somewhat surprising result that T also coincides with the C∗-algebra generated by the class of weakly localized operators.