Duality in Gerstenhaber algebras

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Let C be a differential graded coalgebra, ΩC the Adams cobar construction 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 (ΩC; ΩC). This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.