 for the associative operad was given in [@Tr] and the starting point for a similar version for the commutative operad was considered in [@Ginot]. It is natural to ask for a generalization of these constructions applicable to any cyclic operad. This is what is done in this paper.

Starting with a cyclic operad $\mathcal O$, we use the notion of colored operads to incorporate the cyclic structure of $\mathcal O$ into the colored operad $\widehat{\mathcal O}$. Algebras over the colored operad $\widehat{\mathcal O}$ consist of pairs $(A,M)$, where $A$ is an algebra over $\mathcal O$ and $M$ is an $\mathcal O$-module over $A$ which has an invariant inner product. Section \[cyclic-op\] is devoted to explicitly defining $\widehat{\mathcal O}$, and in the case that $\mathcal O$ is given by quadratic generators and relations, we give a description of $\widehat{\mathcal O}$ in terms of generators and relations coming from those of $\mathcal O$.

A major tool in the theory of operads is the notion of Koszul duality. Let us recall, that a (colored) operad $\mathcal P$ is called Koszul if there is a quasi-isomorphism of operads $\mathbf{D}(\mathcal P^!)\to \mathcal P$, where $\mathbf{D}(\mathcal P^!)$ denotes the dual operad (in the sense of [@GK (3.2.12)]) on the dual quadratic operad $\mathcal P^!$. This implies that the notion of algebras of $\mathcal P$ has a canonical infinity version given by algebras over $\mathcal P_\infty:=\mathbf{D}(\mathcal P^!)$. Our main theorem states, that the Koszulness property is preserved when going from $\mathcal O$ to $\widehat{\mathcal O}$.

Let $\mathcal O$ be a cyclic quadratic operad. If $\mathcal O$ is Koszul, then so is $\widehat{\mathcal O}$, i.e. we have a resolution $\widehat{\mathcal O}_\infty:=\mathbf{D}(\widehat{\mathcal O^!})$ of $\widehat{\mathcal O}$.

The proof of this theorem will be given in section \[quadrat-koszul\]. Theorem \[O\_hat\_Koszul\] justifies the concept of the infinity version of algebras and modules with invariant inner products over cyclic operads $\math