Spectral Triples

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.