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?



Does an H-space have at most one delooping?
Another example is $S^3$. If I am not mistaken, there are exactly $12$ H-space structures on $S^3$. Indeed, we can consider the long exact sequence
$$[S^4\vee S^4, S^3] \to [S^6, S^3] \to [S^3\times S^3, S^3] \to [S^3\vee S^3, S^3]$$
of groups (using an arbitrary loop structure on $S^3$). We have to count the preimages of the folding map $S^3 \vee S^3 \to S^3$ in $[S^3\times S^3, S^3]$. The group $[S^6, S^3] = \pi\_6S^3$ is $\mathbb{Z}/12$. Moreover, the map $S^6 \to S^4\vee S^4$ is null as it is the suspension of the Whitehead product of the two inclusions $S^3 \to S^3\vee S^3$. Thus, the number of $H$-space structures (up to homotopy) is in bijection with $\mathbb{Z}/12$. 


On the other hand, there are uncountably many loop structures on $S^3$. This was proven by Rector in *Loop structures on the homotopy type of $S^3$*. What he does is roughly the following: He considers spaces that are *$p$-locally* equivalent to $\mathbb{HP}^\infty$ for every prime $p$, but not necessarily equivalent to $\mathbb{HP}^\infty$ themselves, and whose loop space is equivalent to $S^3$. He shows that there is a $\{0,1\}^{\infty}$ worth of them, i.e. uncountably many. In particular, there are $H$-space structures on $S^3$ with infinitely many deloopings. 


This technique having certain fixed pieces at all the primes, but mixing them in a new way to obtain a new (integral) space is sometimes called Zabrodsky mixing. There is an amusing description on p.79 of Adams's *Infinite loop spaces*. 


