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 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.

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 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.