A double commutant relation in the Calkin algebra on the Bergman space

Let T be the Toeplitz algebra on the Bergman space La2(B,dv) of the unit ball in Cn. We show that the image of T in the Calkin algebra satisfies the double commutant relation: π(T)={π(T)}″. This is a surprising result, for it is the opposite of what happens in the Hardy-space case [16], [17].