Adams operations, algebras up to homotopy and cyclic homology

We define and study a new A∞-algebra structure on the cyclic bar complex of a commutative algebra. Unlike the Getzler–Jones’ one, this new structure is compatible with the Adams operations and provides therefore the right setting for studying operations on cyclic homology theories. Its construction relies on the properties of some formal polytopes, which should be thought of as intermediates between Stasheff polytopes and models for cyclic homology operations.