Noncommutative integration and spatial theory

This course develops integration on spaces that ordinary measure theory cannot see, such as the space of orbits of an ergodic group action or the leaf space of a foliation. It starts with the spatial derivative, which compares a weight on a von Neumann algebra with a weight on its commutant, and then builds transverse measures on measured groupoids, the von Neumann algebra of random operators, and the weights, formal dimension and index that come with them. Basic references are [Connes 1979], [Connes 1980a], [Connes 1982] and [Connes 1994].

1. The spatial derivative
2. Measured groupoids and transverse measures
3. Almost homomorphisms of measured groupoids
4. Square-integrable representations and random operators
5. Weights on random operators and formal dimension

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