Rawnsley, John On traces for differential star products on symplectic manifolds. (English) Zbl 1041.53056 Voronov, Theodore (ed.), Quantization, Poisson brackets and beyond. London Mathematical Society regional meeting and workshop on quantization, deformations, and new homological and categorical methods in mathematical physics, Manchester, UK, July 6–13, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3201-8). Contemp. Math. 315, 39-45 (2002). Using Karabegov’s uniqueness theorem for normalized traces on open balls in \({\mathbb R}^{2n}\) [A.V. Karabegov, Lett. Math. Phys. 45, No. 3, 217–228 (1998; Zbl 0943.53052)], the author gives a new proof of the Fedosov and Nest-Tsygan theorem on the existence and uniqueness of traces for star-products on symplectic manifolds. The proof goes as follows: Any symplectic manifold \((M,\omega)\) can be covered by Darboux charts which are diffeomorphic to open balls in \({\mathbb R}^{2n}\) such that their intersections are also diffeomorphic to open balls. Any star-product on \(M\) induces by restriction star-products on the charts. These local star-products have normalized traces; by Karabegov’s uniqueness theorem, these local traces glue together to give a global trace.For the entire collection see [Zbl 1007.53002]. Reviewer: Domenico Fiorenza (Roma) MSC: 53D55 Deformation quantization, star products Keywords:deformation quantization; star products; traces Citations:Zbl 0943.53052 × Cite Format Result Cite Review PDF