  In this paper we show that a strongly homotopy commutative (or $C_\infty$-)
algebra with an invariant inner product on its cohomology can be uniquely
extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a
commutative Frobenius algebra introduced by Kontsevich). This result relies on
the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a
$\ci$-algebra and does not generalize to algebras over other operads.
