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$\newcommand{\C}{\mathbb{C}}$ $\newcommand{\P}{\mathbb{P}}$ Let the isomorphism $\C^n \otimes \C^m \to \C^{nm}$ be given by the diagonal ordering $e_0\otimes f_0,e_1 \otimes f_0, e_0 \otimes f_1, e_2 \otimes f_0,...$

This gives us a multiplication $\times$ on $\P^\infty$, $[\underbrace{l_1}_{\in \C^n}]\times [\underbrace{l_2}_{\in \C^m}]=[\underbrace{l_1 \otimes l_2}_{\C^{nm}}]$.

Points in $\P^\infty$ are also loops in $K(\mathbb Z,3)$ which gives another multiplication $\cdot$ via concatenation of loops.

I want to show that the maps $(a \cdot b) \times (c \cdot d) =(a \times c) \cdot (b \times d)$ for any lines $a,b,c,d$ in $\C^\infty$, up to homotopy.

An explicit map from $\P^\infty$ to $\Omega K(\mathbb Z,3)$ would help alot.

Here is one place where I have used this argument before, https://math.stackexchange.com/a/1859287/305314 .

user062295
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  • I think your $\times$ corresponds to the tensor product of line bundles and your $\cdot$ corresponds to the addition in $H^2(-,\mathbb{Z})$. The map $\mathbb{C}P^\infty \to K(\mathbb{Z},2)$ you are looking for is represented by $c_1 \in [\mathbb{C}P^\infty, K(\mathbb{Z},2)] \cong H^2(\mathbb{C}P^\infty, \mathbb{Z})$. So showing that the two multiplications are homotopic is just be the identity $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2)$ of Chern classes. – JHF May 28 '17 at 15:58
  • I have been thinking about your comment for a while now. From what I see you have given me a group isomorphism for every topological space $B$, $([B, \mathbb{CP}^\infty], \otimes) \cong ([B, \mathbb{CP}^\infty], loopspacemultiplication)$. I am not sure how to show this implies that there is an $H$-space isomorphism $\mathbb{CP}^\infty \cong \mathbb{CP}^\infty$.

    I don't think that taking $B=\mathbb{CP}^\infty$ does the trick.

     – user062295 Jun 08 '17 at 17:28

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