This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway: 


Let $G$ be a simply-connected topological group. In particular, it is an $H$-space. Therefore the theorem of Cartan and Serre tells us that the rationalization $G\_{\mathbb{Q}}$ of $G$ has the homotopy type of a product of Eilenberg-Maclane spaces. Here are my questions:



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> > * Can one choose a functorial rationalization $G \to G\_{\mathbb{Q}}$ that also preserves products, thereby turning $G\_{\mathbb{Q}}$ into a topological group and $G \to G\_{\mathbb{Q}}$ into a group homomorphism? Would this group structure coincide up to homotopy with the $H$-space structure on the product of EM-spaces?
> > * If the above is true, then the map $(BG)\_{\mathbb{Q}}\to B(G\_{\mathbb{Q}})$ induced by $G \to G\_{\mathbb{Q}}$ should be a homotopy equivalence. Would this imply that $(BG)\_{\mathbb{Q}}$ also decomposes into a product of Eilenberg-Maclane spaces?
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rationalization of classifying spaces
In the paper


Arkowitz, Martin [*Categories equivalent to the category of rational H-spaces*](http://link.springer.com/article/10.1007/BF01170937), Manuscripta Math. 64 (1989), no. 4, 419–429


it is shown that the rational homotopy equivalence $G\_\mathbb{Q}\cong\prod\_n K(\pi\_n(G\_\mathbb{Q}),n)$ is an equivalence in the category of rational $H$-spaces if and only if $G$ is homotopy abelian (see Proposition 3.1). This means, I think, that the answer to your first question is no in general.


However I think the answer to your second question should be yes, but for different reasons. If $G$ is a connected topological group of the homotopy type of a finite CW complex, then theorems of Hopf and Borel imply that $$H^\ast(BG;\mathbb{Q})\cong\mathbb{Q}[y\_1,\ldots , y\_k]$$ where each generator $y\_i$ is of even degree (see McCleary's *A User's Guide to Spectral Sequences*, Theorem 6.38). It follows that the minimal model of $BG$ is evenly generated, and has zero differential, meaning that $BG$ is rationally a product of Eilenberg-Mac Lane spaces.


