It is well know that the $\infty$-category of group-like $E\_\infty$-spaces and the $\infty$-category of connective spectra are equivalent, see e.g. 
May - "$E\_\infty$-spaces, group completions and permutative categories" or
Lurie - "Higher Algebra", Remark 5.1.3.17


Now the category of $E\_\infty$-spaces (here space means simplicial set) carries a model structure as well as the category of spectra. Is there a direct (left) Quillen functor 


$E\_\infty$-space $\to$ Spectra


whose derived functor restricts to such an equivalence? I have been unable to find a discussion of this in the litertatur. The only thing I can find are indirect functors going through $\Gamma$-spaces or related categories. The Bar-construction which is usually used is not left Quillen (!?).



Equivalence between $E_\infty$-spaces and connective spectra
Of course, as several people have noted, the answer depends on
the choice of details. There is a variant of my original passage
from $E\_{\infty}$ spaces to spectra that certainly works, as was
noted in ``Units of ring spectra and Thom spectra'' by Ando, 
Blumberg, Gepner, Hopkins, and Rezk (arXiv: 0810.4535v3). 


Take the Steiner $E\_{\infty}$ operad for definiteness and 
denote the monad on based spaces associated to it by $\mathbf{C}$.
Take spectra to mean Lewis-May spectra since it is very convenient
to have the $(\Sigma^{\infty},\Omega^{\infty})$ adjunction for the
question at hand, and that is incompatible with symmetric monoidal
categories of spectra. Of course, that means I'm not using simplicial
sets, but I don't suffer from a prejudice in their favor: when I write
space I prefer to actually mean space. 


Then, as discussed in modern terms in my 
paper "What precisely are $E\_{\infty}$ ring spaces and $E\_{\infty}$ 
ring spectra?'' Geometry & Topology Monographs 16(2009), 215--282,
the spectrum associated to a $\mathbf{C}$-space $X$ is the two-sided
bar construction $B(\Sigma^{\infty},\mathbf{C},X)$. For cofibrant 
$X$, this is equivalent to the ``tensor product''  

$\Sigma^{\infty}\otimes\_{\mathbf{C}}X$, which is defined
by an obvious coequalizer. This functor from $\mathbf{C}$-spaces to
spectra is left adjoint to $\Omega^{\infty}$. Further details are as
one would expect.


