A double commutant theorem for Murray–von Neumann algebras

Murray-von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra A of the Murray-von Neumann algebra A(f)(R) associated with a finite von Neumann algebra R is the Murray-von Neumann algebra A(f)(A(0)), where A(0) is a maximal abelian self-adjoint subalgebra of R and, in addition, A(0) is A Π R. We also prove that the Murray-von Neumann algebra A(f)(C) with C the center of R is the center of the Murray-von Neumann algebra A(f)(R). Von Neumann's celebrated double commutant theorem characterizes von Neumann algebras R as those for which R'' = R, where R', the commutant of R, is the set of bounded operators on the Hilbert space that commute with all operators in R. At the end of this article, we present a double commutant theorem for Murray-von Neumann algebras.