Weak operad and deloopings Let $E$ be an operad in topological spaces. $E$ is usually called an $E_{\infty}$-operad, if all the spaces $E_n$ are contracticle. If $E$ acts on a space $X$, then by the recognition principle, $X$ turns out to be an infinite loop space. 

How much of the theory is preserved, if I replace contractible by weakly contractible? Do I still have deloopings, if $E$ acts on a space?

 A: To answer your question and complete Justin's answer, one can pick a cofibrant replacement of $E$ to get a genuine $E_\infty$-operad $F$ acting on the space $X$, and conclude that $X$ has an infinite delooping up to group completion issues.
But the other way round fails: not all infinite loop spaces are acted on by $E$. Example: any group like commutative monoid is an infinite loop space, but not all infinite loop spaces are commutative monoids (otherwise no non-trivial Dyer Lashof operation would exist).
Remark: If $E$ is weakly-contractible, then the constant maps $c: E_n\rightarrow pt$ define an acyclic fibration, in the category of topological operads, from the operad $E$ towards the operad of commutative monoids $C$.
If you have a preferred (cofibrant) $E_\infty$-operad $F$, then you can use the LLP to get an operad weak-equivalence $f: F\xrightarrow{\sim} E$, making $F$ a cofibrant replacement of $E$.
A: Some clarifications: 
1) You need that $X$ is grouplike (so the induced multiplication makes $\pi_0 X$ a group). This condition is always satisfied for a loop space, but not satisfied by the discrete $E_\infty$ space $\mathbb{N}$.
2) In order for $\mathcal{L}$ to be an $E_\infty$ operad we require more than  $\mathcal{L}(n)\simeq *$, otherwise we could have just used the commutative operad and all grouplike $E_\infty$ spaces would be topological abelian groups. We also require that $\mathcal{L}(n)$ is a cofibrant $\Sigma_n$-space, so that it has a free $\Sigma_n$ action and consequently homotopy equivalent to $E\Sigma_n$. 
In May's 'Geometry of iterated loop spaces' especially Ch. 3  http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf  it is shown that the delooping machine does not depend on the choice of $E_\infty$ operad.
