Localization and the Toeplitz algebra on the Bergman space

Let T_f denote the Toeplitz operator with symbol function f on the Bergman space La2(B,dv) of the unit ball in Cⁿ. 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(La2(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.