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

Abstract

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).