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. 7

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