rationalization of classifying spaces 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? 
    
  

 A: In the paper
Arkowitz, Martin Categories equivalent to the category of rational H-spaces, 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.
A: This is a longish comment on James Schwass's answer, not an answer to the 
original questions. 
Have to be a little careful here.  We are deliberately informal (p.97), 
but we are working in a category of "cocellular" spaces and maps 
on which localizations can be constructed to be product-preserving
on the point-set level.  Every nilpotent space $X$ "admits" such a 
Postnikov tower $Y$, but that only gives you a weak equivalence $X\to Y$.
Our arguments give a more precise than usual version of the fact that
rationalization commutes with products up to homotopy: cocellular spaces
admit cocellular localizations; products and more generally pullbacks of 
cocellular localizations are cocellular localizations.  Incidentally,
Bousfield and Kan have nothing to do with this.  On nilpotent spaces,
localizations are the same up to homotopy no matter what construction
one uses.  Bousfield-Kan gives a particular simplicial construction on 
the point-set level, but that construction is not used in May-Ponto.
Note that on non-nilpotent spaces there are several different notions of 
localization (and in particular rationalization), none well understood
calculationally.
