Index theory for locally compact noncommutative geometries. (English) Zbl 1314.46081
Mem. Am. Math. Soc. 1085, v, 127 p. (2014).
Summary: 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. 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.
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. 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.
MSC:
| 46L87 | Noncommutative differential geometry |
| 46L51 | Noncommutative measure and integration |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 19K35 | Kasparov theory (\(KK\)-theory) |
| 19K56 | Index theory |
| 58J05 | Elliptic equations on manifolds, general theory |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 58J30 | Spectral flows |
| 58J32 | Boundary value problems on manifolds |
| 58J42 | Noncommutative global analysis, noncommutative residues |
Keywords:
local index formula; spectral triple; Fredholm module; Kasparov product; nonunital algebras; Gromov-Lawson relative index formula; McKean-Singer formulaReferences:
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