Von Neumann Algebra - Modules Over A Factor

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_M(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 = 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_n (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 n2.)
• Type I_∞ 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 II₁: The M-dimension can be anything in . 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_∞: The M-dimension can be anything in . 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.