Milnor idempotents through Toeplitz operators

Geometric generators of the even  topological K-theory of a projective space were constructed by Adams. There is also known a noncommutative deformation (aka quantization) of a projective space under which  the K-theory doesn't change. We construct noncommutative counterparts of Adams generators in terms of noncommutative associated vector bundles and a noncommutative version of the Milnor formula related to the Heegard splitting. Then we construct a homotopy between the topological Milnor idempotent and a Toeplitz type projection. This can be viewed, in the spirit of the Atiyah-Singer theorem, as an interpolation between the topology of a manifold and operator theory, in which the role of an elliptic operator is played by quantization.

(joint with  Carla Farsi, Piotr M. Hajac,and Bartosz Zielinski)