Affiliated subspaces and the structure of von neumann algebras

The interplay between order-theoretic properties of structures of subspaces affiliated with a von Neumann algebra M and the inner structure of the algebra M is studied. The following characterization of finiteness is given: a von Neumann algebra M is finite if and only if in each representation space of M one has that closed affiliated subspaces are given precisely by strongly closed left ideals in M. Moreover, it is shown that if the modular operator of a faithful normal state φ is bounded, then all important classes of affiliated subspaces in the GNS representation space of φ coincide. Orthogonally closed affiliated subspaces are characterized in terms of the supports of normal func-tionals. It is proved that complete affiliated subspaces correspond to left ideals generated by finite sums of orthogonal atomic projections. © Theta, 2013