Noncommutative Geometry Seminar

If the quotient of a compact Hausdorff space by a suitable group action is again a compact Hausdorff space, then the C*-algebra of the quotient is simply the fixed point subalgebra of the C*-algebra of the original space. If the quotient of a compact oriented Riemannian manifold by a suitable Lie group action is again a compact oriented Riemannian manifold, what happens at the level of spectral triples? In this talk, I will discuss what it means for a compact Lie group action on a spectral triple to admit a good quotient in the form of a spectral triple, and I will give an unbounded KK-theoretic construction of a good quotient for the commutative spectral triple corresponding to a generalised Dirac operator equivariant under a free and isometric action of a compact connected Lie group. As time permits, I will then discuss applications to noncommutative principal bundles arising via Rieffel's strict deformation quantisation. This is joint work with Bram Mesland.