CHAPTER 1
Pseudodifferential Calculus and Summability
In this Chapter we introduce our chief technical innovation on which most of
our results rely. It consists of an L1-type summability theory for weights adapted
to both the nonunital and noncommutative settings.
It has become apparent to us while writing, that the integration theory pre-
sented here is closely related to Haagerup’s noncommutative Lp-spaces for weights,
at least for p = 1, 2. Despite this, it is sufficiently different to require a self-
contained discussion.
It is an essential and important feature in all that follows that our approach
comes essentially from an
L2-theory:
we are forced to employ weights, and a direct
L1-approach
is technically unsatisfactory for weights. This is because given a weight
ϕ on a von Neumann algebra, the map T ϕ(|T |) is not subadditive in general.
Throughout this chapter, H denotes a separable Hilbert space, N B(H) is a
semifinite von Neumann algebra, D : dom D H is a self-adjoint operator affiliated
to N , and τ is a faithful, normal, semifinite trace on N . Our integration theory
will also be parameterised by a real number p 1, which will play the role of a
dimension.
Different parts of the integration and pseudodifferential theory which we in-
troduce rely on different parts of the above data. The pseudodifferential calculus
can be formulated for any unbounded self-adjoint operator D on a Hilbert space H.
This point of view is implicit in Higson’s abstract differential algebras, [32], and
was made more explicit in [15].
The definition of summability we employ depends on all the data above, namely
D, the pair (N , τ) and the number p 1. We show in Section 1.1 how the pseu-
dodifferential calculus is compatible with our definition of summability for spectral
triples, and this will dictate our generalisation of a finitely summable spectral triple
to the nonunital case in Chapter 2.
The proof of the local index formula that we use in the nonunital setting requires
some estimates on trace norms that are different from those used in the unital case.
These are found in Section 1.5. To prepare for these estimates, we also need some
refinements of the pseudodifferential calculus introduced by Connes and Moscovici
for unital spectral triples in [22,25].
1.1. Square-summability from weight domains
In this Section we show how an unbounded self-adjoint operator affiliated to a
semifinite von Neumann algebra provides the foundation of an integration theory
suitable for discussing finite summability for spectral triples.
Throughout this Section, we let D be a self-adjoint operator affiliated to a
semifinite von Neumann algebra N with faithful normal semifinite trace τ, and let
p 1 be a real number.
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