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(H) to the Calkin algebra B(H)/K(H), then π(A)≠π(B) and {π(A)}″=π(B).