Reflexivity Of Murray-Von Neumann Algebras

A Murray-von Neumann algebra Af (R) is the algebra of operators affiliated with a finite von Neumann algebra R. Such an algebra contains both bounded and unbounded operators on a Hilbert space. In this article, we study reflexivity of Murray-von Neumann algebras. We discuss the stability of closed subspaces of a Hilbert space H under closed, densely defined operators on H, based on which we define ̂LatS of a set S of closed, densely defined operators on H, and ÂlgP of a set P of closed subspaces of H. We show that Murray-von Neumann algebras Af (R) are reflexive, that is, Af (R) ≌ÂlĝLatAf (R). We also define ̂RefaAf (R), and show that Murray-von Neumann algebras Af (R) are algebraically reflexive, that is, Af (R) ≌̂RefaAf (R)