TITLE: Does an H-space have at most one delooping?
QUESTION [14 upvotes]: I am new to H-spaces, delooping, etc. I know that not every $H$-space has a delooping (e.g. Stasheff's theorem, one needs a group-like $A_\infty$ space).  I also know that the same space can have inequivalent deloopings corresponding to different $H$-space structures.  An example is $BU$, which has two $H$-space structures (coming from direct sum, respectively tensor product, of vector bundles).  
Once you fix an $H$-space structure on a space $X$, is there at most one (up to homotopy) delooping $BX$ such that $X$ and $\Omega BX$ are equivalent as $H$-spaces, in some appropriate sense?

REPLY [10 votes]: The real projective spaces $\mathbb{R}P^3 \cong SO(3)$ and $\mathbb{R}P^7$ also give fun examples. Naylor proved that there exist 768 $H$-space structures on $SO(3)$, while Rees shows that there exist 30,720 (!) $H$-space structures on $\mathbb{R}P^7$.
For the spheres, James proved that $H$-space structures on $S^n$ (when they exist) are in 1-1 correspondence with elements of $\pi_{2n}S^n$. When $n = 3$, this gives Lennart's calculation that there exist 12 $H$-space structures on $S^3$. When $n = 7$, we have $\pi_{14}S^7 \cong \mathbb{Z}/120$, so there exist 120 $H$-space structures on $S^7$.