Editing Von Neumann algebra (section)

==Weights, states, and traces==
{{Further|Noncommutative measure and integration}}

Weights and their special cases states and traces are discussed in detail in {{harv|Takesaki|1979}}.

*A '''weight''' ω on a von Neumann algebra is a linear map from the set of [[C*-algebra#Self-adjoint elements|positive elements]] (those of the form ''a*a'') to [0,∞].
*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 (functional analysis)|state]]''' is a weight with ω(1)&nbsp;=&nbsp;1.
*A '''trace''' is a weight with ω(''aa*'')&nbsp;=&nbsp;ω(''a*a'') for all ''a''.
*A '''tracial state''' is a trace with ω(1)&nbsp;=&nbsp;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 over the projections of the factor, as follows:
*Type I<sub>''n''</sub>: 0, ''x'', 2''x'', ....,''nx'' for some positive ''x'' (usually normalized to be 1/''n'' or 1).
*Type I<sub>∞</sub>: 0, ''x'', 2''x'', ....,∞ for some positive ''x'' (usually normalized to be  1).
*Type II<sub>1</sub>: [0,''x''] for some positive ''x'' (usually normalized to be  1).
*Type II<sub>∞</sub>: [0,∞].
*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.