In his paper Cyclic homology, Derivations and the Free Loopsace, Goodwillie defines periodic cyclic homology for differential graded algebras (A,d) concentrated in non-negative degree. Why does he make this restriction?
Elsewhere, for example in Jones' work on the same topic, periodic cyclic homology is defined for arbitrary dga's. Suppose that we have a dg-algebra which is co-connective, e.g. $A_n=0$ for $n>0$.
Is it true that $\operatorname{HP}_*((A,d)) \cong \operatorname{HP}_*(H_0(A))$ ?