On the essential commutant of the Toeplitz algebra on the Bergman space

Let T be the C⁎-algebra generated by the Toeplitz operators {Tf:f∈L∞(B,dv)} on the Bergman space of the unit ball. We show that the essential commutant of T equals {Tg:g∈VObdd}+K, where VObdd is the collection of bounded functions of vanishing oscillation on B and K denotes the collection of compact operators on La2(B,dv).