Notes on Kontsevich-Soibelman's theorem about cyclic A-infinity algebras

Kontsevich and Soibelman has proved a relation between a non-degenerate cyclic homology element of an A-infinity algebra A and its cyclic inner products on the minimal model of A. We find an explicit formula of this correspondence, in terms of the strong homotopy inner products and negative cyclic cohomology of A. We prove that an equivalence class of the induced strong homotopy inner product depends only on the given negative cyclic cohomology class. Also, we extend such a correspondence to the case of gapped filtered A-infinity algebras.