Suppose I have an operad $P(n)$ in topological spaces in the sense of the book by Markl, Shnider and Stasheff, i.e. there is *no* space $P(0)$. In agreement with some of the answers to  [this](https://mathoverflow.net/questions/29636/operad-terminology-operads-with-and-without-o0) question, I will call such an operad *unreduced*. 


Let $H$ be an infinite dimensional separable Hilbert space. Then an example for $P$ would be 


$$
P(n) = Iso(H^{\otimes n}, H)
$$


the linear isometric isomorphisms from $H^{\otimes n}$ to $H$. There is no obvious candidate for $P(0)$ (since we insist on isomorphisms, $P(0) = Hom(\mathbb{C},H)$ will not work). 


It is still possible to talk about unreduced $E\_{\infty}$-operads in this case. But now it is less clear (at least to me) that I can deloop a space, which is an algebra over an unreduced $E\_{\infty}$-operad. Hence my question:



> 
> Let $X$ be a space, which is an algebra over an unreduced $E\_{\infty}$-operad $P$, such that $\pi\_0(X)$ is a group. Is $X$ an infinite loop space?
> 
> 
> 


If this is not true:



> 
> Are there conditions on the space or the operad (other than that it can be reduced) to ensure that $X$ is an infinite loop space?
> 
> 
> 


Using the bar construction by May, I think this would correspond to the question, whether I can drop degeneracies and still get a delooping. 



Delooping and unreduced operads
The answer to your first question is no: for example, take $X$ to be any connected pointed space, and regard $X$ as a nonunital commutative monoid by saying that the product of
any two points of $X$ is equal to the base point. This endows $X$ with the structure of
an algebra over any operad $\mathcal{O}$ with $\mathcal{O}(0) = \emptyset$, but $X$ need not be an infinite loop space.


However, with a slightly stronger hypothesis, the answer is yes. Let $X$ be a nonunital
$E\_{\infty}$-space with the property that translation by any point $x \in X$ is a weak homotopy equivalence from $X$ to itself. Then $X$ is weakly homotopy equivalent to an infinite loop space.


Assuming that your operads are sufficiently nice (so that "algebras up to coherent homotopy" can be rectified), one has the following more precise statement: the homotopy category of $E\_{\infty}$-spaces is equivalent to the subcategory of
the homotopy category of "nonunital" $E\_{\infty}$-spaces, whose objects are nonunital
$E\_{\infty}$-spaces $X$ for which $\pi\_0 X$ contains a unit element $e\_X$ such that translation by (any representative of) $e\_X$ is a weak homotopy equivalence, and whose morphisms are maps of nonunital $E\_{\infty}$-spaces $X \rightarrow Y$ which carry
$e\_{X}$ to $e\_{Y}$.


