Let $M$ be a finitely generated free $\mathbb{Z}$-module of rank $n$. Consider the associated group algebra $K[M]$ and the algebraic torus $\mathbb{T}_{M}=\operatorname{Spec}(K[M])$. The corresponding analytic space $\mathbb{T}_{M}^{{\text{\rm an}}}$ is the set of multiplicative seminorms of $K[M]$ that extend the absolute value of $K$. This is an analytic group. We warn the reader that the set of points of an analytic group is not an abstract group, hence some care has to be taken when speaking of actions and orbits. The precise definitions and basic properties can be found in [Ber90, §5.1].

Its analytification $\mathbb{T}_{M}^{{\text{\rm an}}}$ is an analytic torus as in [Ber90, §6.3]. The subset

is a compact subgroup, called the compact torus of $\mathbb{T}_{M}^{{\text{\rm an}}}$.