Non-Commutative Integration

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The theories of weights, traces and states are often referred as non commutative integration. If the von Neumann algebra in question is abelian, then our theory is precisely the theory of measures and integration. In fact, the weight value of a self-adjoint element is given precisely by the integration of the corresponding function on the spectrum relative to the measure corresponding to the weight. As there are many non-commuting self-adjoint elements in the algebra, we have to consider various spectral measures even if we fix one weight and we can not represent non-commuting self-adjoint elements as functions on the same space. The striking difference between the commutative case and the non-commutative case is the appearance of one parameter automorphism group which is determined by the weight. Namely, weights and/or states determine the dynamics of the system which does not have the commutative counter part. We have explored the relationship between weights and the modular automorphism groups so far. We now further investigate how the dynamics, i.e. the modular automorphism groups, of the algebra relate the different spaces associated with the algebras. First, we study the underlying Hilbert space of the algebra and find the intrinsic pointed convex cone there, which is called the natural cone, in the first section. The theory developed there allows us to view the standard Hilbert space as the square root of the predual of the algebra as well as to represent the automorphism group Aut(M), of a von Neumann algebra M, as the group of unitaries which leaves the natural cone globally invariant.