Cyclic $A_\infty$-algebras and cyclic homology

We provide a new description of the complex computing the Hochschild homology of an -unitary -algebra as a derived tensor product such that: (1) there is a canonical morphism from it to the complex computing the cyclic homology of that was introduced by Kontsevich and Soibelman, (2) this morphism induces the map in the well-known SBI sequence, and (3) is canonically isomorphic to the space of morphisms from to in the derived category of -bimodules. As direct consequences we obtain previous results of Cho and Cho–Lee, as well as the fact that Koszul duality establishes a bijection between (resp., almost exact) -Calabi–Yau structures and (resp., strong) homotopy inner products, extending a result proved by Van den Bergh.