Coincidence of essential commutant and the double commutant relation in the Calkin algebra

Let B be a von Neumann algebra on a separable Hilbert space H. We show that, if the dimension of B as a linear space is infinite, then it has a proper C*-subalgebra A whose essential commutant in ℬ(H) coincides with the essential commutant of B. Moreover, if π is the quotient map from ℬ B(H) to the Calkin algebra ℬ (H)/K(H), then π(A) ≠ π(B) and {π(A)}″ = π(B).