Deformation quantization with traces

In the present paper we prove a statement closely related to the cyclic formality conjecture [Sh]. In particular, we prove that for a volume form Ω and a Poisson bivector field π on Rd such that divΩ π = 0, the Kontsevich star-product [K] with the harmonic angle function is cyclic, i.e. ∫ Rd (f ∗ g) · h · Ω = ∫ Rd (g ∗ h) · f · Ω for any three functions f, g, h on Rd (for which the integrals make a sense). We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and prove a generalization of the Connes-Flato-Sternheimer conjecture [CFS] on closed star-products in the Poisson case. 1 Cyclic formality conjecture We work with the algebra A = C(M) of smooth functions on a smooth manifold M . One associates to the algebra A two differential graded Lie algebras: the Lie algebra T • poly(M) of smooth polyvector fields on the manifold M (with zero differential and the Schouten-Nijenhuis bracket), and the polydifferential part D poly(M) of the cohomological Hochschild complex of the algebra A, equipped with the Gerstenhaber bracket (see [K] for the definitions). We consider T • poly(M) and D • poly(M) to be graded as Lie algebras, i.e. T i poly(M) = {(i+ 1)-polyvector fields} and D i poly(M) ⊂ HomC (A , A). The formality theorem of Maxim Kontsevich [K] states that T • poly(M) and D poly(M) are quasi-isomorphic as differential graded (dg) Lie algebras, i.e.