Traces for star products on symplectic manifolds. (English) Zbl 1075.53097

Let \(\ast\) be a differential star product on a symplectic manifold \((M, \omega)\). A trace is a (non trivial) \({\mathbb C}[[\nu]]\)-linear map \(\tau : C^{\infty}_c(M)[[\nu ]] \rightarrow {\mathbb C}[\nu ^{-1},\nu]]\) satisfying \(\tau (f\ast g)=\tau (g\ast f)\). It is known that such a trace exists and is unique up to multiplication by an element of \({\mathbb C}[\nu ^{-1},\nu]]\) [B. V. Fedosov, Sov. Phys., Dokl. 31, 877–878 (1986); translation from Dokl. Akad. Nauk SSSR 291, 82–86 (1986; Zbl 0635.58019), Deformation quantization and index theory, Mathematical Topics. 9. Berlin: Akademie Verlag (1996; Zbl 0867.58061), R. Nest and B. Tsygan, Commun. Math. Phys. 172, No. 2, 223–262 (1995; Zbl 0887.58050)]. The authors give a direct elementary proof of this result by using some ideas and techniques from their own papers [J. Geom. Phys. 29, No. 4, 347–392 (1999; Zbl 1024.53057)] and A. V. Karabegov [Lett. Math. Phys. 45, No. 3, 217–228 (1998; Zbl 0943.53052)].


53D55 Deformation quantization, star products
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