I am interested in infinite loop structures on the infinite dimensional projective space $\mathbb{R} P^\infty$. Is it unique? I think this has to be known in work of May, and If so, then I presume its proof should boil down to some simple fact. Can anyone please give me some advise on this, and the simple fact that one might expect?! 



Loop space structures on $RP^\infty$
Essentially by definition, the category of infinite loop spaces is equivalent to the category of $(-1)$-connected spectra. Thus, you are asking about $(-1)$-connected spectra $X$ such that there exists an equivalence $\Omega^\infty X\simeq\mathbb{R}P^\infty$ of spaces (ignoring any infinite loop structures). From such an equivalence we see that $\pi\_\*X$ is the same as $\pi\_\*\mathbb{R}P^\infty$, and so consists of a single $\mathbb{Z}/2$ in degree $1$. Thus, there is only one possibility for $X$, by the standard argument for uniqueness of Eilenberg-MacLane spectra.


