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

摘要

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 pi is the quotient map from B(H) to the Calkin algebra B(H)/K(H), then pi(A) not equal pi(B) and {pi(A)}" = pi(B). (C) 2002 Elsevier Science (USA). All rights reserved. [References: 8]