Localization and the Toeplitz algebra on the Bergman space

Let TfTf denote the Toeplitz operator with symbol function f   on the Bergman space La2(B,dv) of the unit ball in CnCn. It is a natural problem in the theory of Toeplitz operators to determine the norm closure of the set {Tf:f∈L∞(B,dv)}{Tf:f∈L∞(B,dv)} in B(La2(B,dv)). We show that the norm closure of {Tf:f∈L∞(B,dv)}{Tf:f∈L∞(B,dv)} actually coincides with the Toeplitz algebra TT, i.e., the C⁎C⁎-algebra generated by {Tf:f∈L∞(B,dv)}{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 TT also coincides with the C⁎C⁎-algebra generated by the class of weakly localized operators.