In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed subvarieties $T_i \cong \mathbb{G}_m^{r_i}$ isomorphic to algebraic tori such that $\bigsqcup_{i \in I} T_i(K) = X(K)$ for every algebraically closed field $K$. I'm trying to prove that $\Omega_{\overline{T_i}}^1(\text{log}D_i)$, the differential forms on $\overline{T_i}$ with logarithmic poles along $D_i = \overline{T_i} \setminus T_i$, is trivial for each $T_i$. I'm actually not sure if this is true but that's my hope. A similar result is true for toric varieties. If $Y$ is a toric variety with open orbit $B$, then $\Omega_{Y}^1(\text{log}(Y \setminus B))$ is trivial (proven in "Toric varieties" by Fulton, pg 87).

My specific question is this: are the $\overline{T_i}$ in $X$ toric varieties? I know this would be true if the action of $T_i$ on itself extended to an action of $T_i$ on $\overline{T_i}$ but I'm not sure if this is possible in general or if not when this is possible. Thanks so much for the help!

Edit: For completeness, the definition of toric variety I'm using is a variety $X$ with an open dense subset $B$ isomorphic to a torus such that the action of $B$ on itself extends to $X$. Thus since each $T_i$ above is locally closed by assumption, then it is an open dense subsets of $\overline{T_i}$ so I only need to guarantee that the action of $T_i$ on itself extends to the closure.

Edit 2: I'll rephrase my question in a more general way so that it maybe easier to approach: if $X$ and $Y$ are varieties, $\overline{Y}$ is a closure of $Y$ such that the inclusion of $Y$ is an open immersion, and $f:X \times Y \to Y$ is a morphism, under what conditions if any does $f$ extend to a morphism $f_* : X \times \overline{Y} \to \overline{Y}$? Furthermore, does the special case of $X$ and $Y$ being a torus $T$, and $f$ being the map $T \times T \to T, \enspace (t,s) \mapsto ts$ as above satisfy these conditions?