Abstract
The central role of the Dirac operator in the geometry of spin manifolds, illustrated in the previous chapter, reveals the central thesis of noncommutative geometry: that the structures we call geometrical are at the same time, and perhaps more fundamentally, operator-theoretic in nature. The transition to the noncommutative world entails putting the metric-generating operator front and centre. This modern approach to geometry is played out on a stage which is a Hilbert space H, on which act both an algebra A and an operator D; together, they form a spectral triple (A, H , D). Guided in part by index theory, we develop in this chapter the cohomological structure of spectral triples; from that structure there emerge several operatorial properties that allow us to assemble the necessary data for noncommutative geometries.
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© 2001 Springer Science+Business Media New York
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Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H. (2001). Spectral Triples. In: Elements of Noncommutative Geometry. Birkhäuser Advanced Texts. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0005-5_10
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DOI: https://doi.org/10.1007/978-1-4612-0005-5_10
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6569-6
Online ISBN: 978-1-4612-0005-5
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