Deformation quantization with traces

In this Letter we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a constant volume form Omega and a Poisson bivector field pi on R-d such that div(Omega)pi =0, the Konts evich star product with the harmonic angle function is cyclic, i.e. integra l (d)(R)(f*g).h.Omega=integral (d)(R)(g*h).f.Omega for any three functions f,g,h on R-d (for which the integrals make sense). We also prove a globaliz ation of this theorem in the case of arbitrary Poisson manifolds and an arb itrary volume form, and prove a generalization of the Connes-Flato-Sternhei mer conjecture on closed star products in the Poisson case.