Von Neumann Algebra - Weights, States, and Traces

Weights, States, and Traces

Weights and their special cases states and traces are discussed in detail in (Takesaki 1979).

• A weight ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form a*a) to .

• A positive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity).

• A state is a weight with ω(1)=1.

• A trace is a weight with ω(aa*)=ω(a*a) for all a.

• A tracial state is a trace with ω(1)=1.

Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. The type of a factor can be read off from the possible values of this trace as follows:

• Type I_n: 0, x, 2x, ....,nx for some positive x (usually normalized to be 1/n or 1).
• Type I_∞: 0, x, 2x, ....,∞ for some positive x (usually normalized to be 1).
• Type II₁: for some positive x (usually normalized to be 1).
• Type II_∞: .
• Type III: 0,∞.

If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then the functional a → (av,v) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the GNS construction for normal states.