this is a question about the action of the formal tori defined in recent papers of Andreatta, Iovita and Pilloni. The notations are heavy, so I will follow the paper Triple product p-adic L-functions associated to finite slope p-adic families of modular forms. At the end of the proof of Thm. 4.6, when considering the action of the formal tori $I^\mathrm{ext}$ on certain element in $\mathbb{W}$, the authors wrote: by density, we only need to consider the action of $\mathbb{Z}_p^\times$. What is the reason lying behind for the density result? It seems $\mathbb{Z}_p^\times$ is in general not dense in $I^\mathrm{ext}$ in a naive sense. Would anyone help me understand this? Thanks very much.
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