Integration on locally compact noncommutative spaces

Abstract

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two examples in the text.In order to understand what is new in our approach in the commutative setting we prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds our index formula appears to be completely new. As we prove our local index formula in the framework of semifinite noncommutative geometry we are also able to prove, for manifolds of bounded geometry, a version of Atiyah's L 2 -index Theorem for covering spaces. We also explain how to interpret the McKean-Singer formula in the nonunital case.In order to prove the local index formula, we develop an integration theory compatible with a refinement of the existing pseudodifferential calculus for spectral triples. We also clarify some aspects of index theory for nonunital algebras.

Pseudo-Riemannian spectral triples and the harmonic oscillator

“…The interesting situation in this setting is when the underlying space is noncompact. In order to discuss summability in this context, we recall a few definitions and results from [CGRS2], where a general definition of summability in the nonunital/noncompact context was developed.…”

“…The space B 2 (D, p) is in fact a Fréchet algebra, [CGRS2,Proposition 2.6], and plays the role of bounded square integrable operators. Next we introduce the bounded integrable operators.…”

“…This is because our definition relies on control of the trace norm of the non-self-adjoint operators a(1 + D 2 ) −s/2 , a ∈ A. It is shown in [CGRS2,Propositions 3.16,3.17] that for a spectral triple (A, H, D) to be finitely summable with spectral dimension p, it is necessary that A ⊂ B 1 (D, p) and this condition is almost sufficient as well. Anticipating the pseudodifferential calculus, we introduce subalgebras of B 1 (D, p) which 'see' smoothness as well as summability.…”

“…In [CGRS2] it was shown that the local index formula holds for smoothly summable spectral triples. Examples show, [CGRS2,page 45], that we can have…”

“…Thus one obtains a K-homology class and the tools to compute index pairings using the local index formula. Since the most important Lorentzian manifolds are noncompact, we have taken care to ensure that our definitions are consistent with the nonunital version of the local index formula, as proved in [CGRS2].…”