Gerstenhaber duality in Hochschild cohomology

Let C be a differential graded chain coalgebra, Ω¯C the reduced cobar construction on C and C∨ the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies HH*(C∨,C∨) and HH*(Ω¯C;Ω¯C). This result yields to a Hodge decomposition of the loop space homology of a closed oriented manifold, when the field of coefficients is of characteristic zero.