I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a number field.

 - **Question 1.** Is the quotient of $\widehat{\mathcal{O}}_K^{\times}$ (the units in the profinite completion of $\mathcal{O}_K$) by the closure of the unit group $\mathcal{O}_K^{\times}$ of $\mathcal{O}_K$ isomorphic, via the Artin map, to the Galois group of the maximal totally real sub-field of the maximal abelian extension $K^{\rm ab}$?
- **Question 2.** Is the Galois group $\text{Gal}(K^{\rm ab}/H)$, for $H$ the Hilbert class field, isomorphic to $\widehat{\mathcal{O}}_K^{\times}$ modulo the closure of the global units? (this would be in conflict with the first question, so I am also implicitly asking how to tweak this statement). Can it be at least written as a quotient of $\widehat{\mathcal{O}}_K^{\times}$ modulo the closure of some subgroup (maybe the closure of $\mathcal{O}_K^{\times,+}$, the group of totally positive units? It would work for $K=\mathbf{Q}$ and imaginary quadratic fields at least) related to $K$ and/or $H$?

As a side question, I know that when $K$ has class number one, the idéle group is $\mathbf{A}_K^{\times} = (K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$. Is there a description of $\mathbf{A}_K^{\times}$ in general, involving $H$? 

**Remark.** On the side question, maybe $\mathbf{A}_K^{\times}$ can be realized as an extension of topological groups, involving the direct product $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$ as the first of the terms? My hope is that one can have, say, $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$ to be the image under the norm map $$\text{Norm}_{H/K}: \mathbf{A}_H^{\times}\to \mathbf{A}_K^{\times}$$ and so $\mathbf{A}_K^{\times}$ can be realized as an extension of the class group by the direct product $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$. I don't know if this is true (is it as a consequence of the Principal Ideal Theorem?), but a statement like this is what I'd be interested in! If true, a reference would be great.