The homotopy invariance of cyclic homology of $A_{\mathrm{\infty }}$-algebras over rings

In the present paper the cyclic homology functor from the category of $A_{\mathrm{\infty }}$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of $A_{\mathrm{\infty }}$-algebras into isomorphisms of graded modules. As a corollary, it is obtained that the cyclic homology of an $A_{\mathrm{\infty }}$-algebra over any field is isomorphic to the cyclic homology of the $A_{\mathrm{\infty }}$-algebra of homologies for the source $A_{\mathrm{\infty }}$-algebra.