Chapter I. Noncommutative Integration
By noncommutative integration, we mean the theory which deals with (semi)cyclic
representations induced by states or weights and states or weights themselves. The most
basic objects in our theory are obviously Hubert spaces and Operator algebras acting on them.
The distinguished property of a Hubert space is of course the inner products. One can view
Hubert spaces as infinite dimensional Euclidean spaces. This view is certainly right as far as
their algebraic properties are concerned. But this view tends to lead us to look at Hubert
spaces as l2 spaces. On the other hand, many Hubert spaces come up along with integration,
for example as L2-spaces. There, the inner product is given by genuine integration. Here the
most important elementary result, in my opinion, is the Riesz representation theorem of lin-
ear functionals. For example, the usual Radon-Nikodym theorem can be proven by making
use of the Riesz theorem. Indeed, it is possible, though not many mathematicians tried it in
their classes, to develop a substantial part of analysis via Hubert space approach, including
integration, distributions and others.
However, an abstract Hubert space alone cannot do much for us. We have to impose
more structure on it. In fact, there is no abstract Hubert space in real life. Every Hubert
space arises through a specific construction. In various ways, the construction of Hubert
space involves integration. But, often the integration directly involved in the construction
does not represent the true picture of our object. One has to reconstruct the entire structure
through an appropriate way—often noncommutative integration. Á typical example of this is
the Fourier transform. Even if the Hubert space is constructed as the usual L2-space, the
appropriate reconstruction often goes beyond the classical frame of integration. One knows
this phenomena well in the type I Situation. Á typical case is the representation theory of
compact groups.
Our claim here is that to every collection of Operators, bounded or unbounded, on a
Hilbert space, there corresponds naturally a "(noncommutative) integration procedure".
Until the Tomita-Takesaki theory was established, noncommutative integration had
been restricted to the theory of traces, although the concept of states was a direct analogue
of probability measures. So the above claim could not have a concrete support since one
could not give manageable contents to the "noncommutative integration procedure". Today
one can handle noncommutative integration more successfully.
Since we are concerned with integration, we consider a von Neumann algebra
, ö} · When we are studying a particular dass of Operators (unbounded), we consider
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