Iterated cyclic homology

From the viewpoint of rational homotopy theory, we introduce an iterated cyclic homology of commutative differential graded algebras over the rational field, which is a generalization of the ordinary cyclic homology of such algebras. Let $\mathbb{T}$ be the circle group and ${\mathcal F}(\mathbb{T}^l, X)$ denote the function space of continuous maps from the $l$-dimensional torus $\mathbb{T}^l$ to an $l$-connected space $X$. It is also shown that the iterated cyclic homology of the differential graded algebra of polynomial forms on $X$ is isomorphic to the rational cohomology algebra of the Borel space $E\mathbb{T}\times_{\mathbb{T}} \mathcal{F}({\mathbb{T}}^l, X)$, where the $\mathbb{T}$-action on ${\mathcal F}(\mathbb{T}^l, X)$ is induced by the diagonal action of $\mathbb{T}$ on the domain $\mathbb{T}^l$.