Recall that one has the equality $\mathbb{A}_f^{\times}=\mathbb{A}^{\times}_{\mathbb{Q},f}=\mathbb{Q}^{\times}.\prod\limits_{p<\infty}\mathbb{Z}_p^{\times}$, so I would like to know if its generalization to arbitrary number field still holds, i.e. let $k$ be a number field, then is it true that $\mathbb{A}^{\times}_{k,f}=k^{\times}.\prod\limits_{v<\infty}\mathcal{O}_v^{\times}$? Here we identify $k^{\times}$ with its image of diagonal embedding in $\mathbb{A}^{\times}_{k,f}$.
Thanks very much!