Editing Von Neumann algebra (section)

==Modules over a factor==
Given an abstract separable factor, one can ask for a classification of its modules, meaning the separable Hilbert spaces that it acts on. The answer is given as follows: every such module ''H'' can be given an ''M''-dimension dim<sub>''M''</sub>(''H'') (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the same ''M''-dimension. The ''M''-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller or equal ''M''-dimension.

A module is called '''standard''' if it has a cyclic separating vector. Each factor has a standard representation, which is unique up to isomorphism. The standard representation has an antilinear involution ''J'' such that ''JMJ'' = ''{{prime|M}}''. For finite factors the standard module is given by the [[GNS construction]] applied to the unique normal tracial state and the ''M''-dimension is normalized so that the standard module has ''M''-dimension 1, while  for infinite factors the standard module is the module with ''M''-dimension equal to ∞.

The possible ''M''-dimensions of modules are given as follows:
*Type I<sub>''n''</sub> (''n'' finite): The ''M''-dimension can be any of 0/''n'', 1/''n'', 2/''n'', 3/''n'', ..., ∞. The standard module has ''M''-dimension 1 (and complex dimension ''n''<sup>2</sup>.)
*Type I<sub>∞</sub> The ''M''-dimension  can be any of 0, 1, 2, 3, ..., ∞. The standard representation of ''B''(''H'') is ''H''⊗''H''; its ''M''-dimension is ∞.
*Type II<sub>1</sub>: The ''M''-dimension can be anything in [0, ∞]. It is normalized so that the standard module has ''M''-dimension 1. The ''M''-dimension is also called the '''coupling constant''' of the module ''H''.
*Type II<sub>∞</sub>: The ''M''-dimension can be anything in [0, ∞]. There is in general no canonical way to normalize it; the factor may have outer automorphisms multiplying the ''M''-dimension by constants. The standard representation is the one with ''M''-dimension ∞.
*Type III: The ''M''-dimension can be 0 or ∞. Any two non-zero modules are isomorphic, and all non-zero modules are standard.