Localization and the Toeplitz algebra on the Bergman space

Abstract

Let T f denote the Toeplitz operator with symbol function f on the Bergman space L a 2 ( B , d v ) of the unit ball in C n . It is a natural problem in the theory of Toeplitz operators to determine the norm closure of the set { T f : f ∈ L ∞ ( B , d v ) } in B ( L a 2 ( B , d v ) ) . We show that the norm closure of { T f : f ∈ L ∞ ( B , d v ) } actually coincides with the Toeplitz algebra T , i.e., the C ⁎ -algebra generated by { T f : f ∈ L ∞ ( B , d v ) } . 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.