Factorial type I KMS states of Lie groups

Motivated by the study of KMS conditions for C*- or W*-dynamical systems defined by covariant unitary representations of topological groups, we consider Gibbs states of a finite-dimensional Lie group $G$ and prove that these are precisely the factorial type I KMS states. For an element $X\in \textbf{L}(G)$ and an irreducible unitary representation $\rho$ of $G$ satisfying $\text{tr}(e^{i\partial\rho(X)})=1$, the corresponding Gibbs state is defined as $\varphi(g)=\text{tr}(\rho(g)e^{i\partial\rho(X)})$. We prove that under the mild assumption that $\rho$ has discrete kernel, the condition $\text{tr}(e^{i\partial\rho(X)})<~\infty$ implies that the generator $X$ is an inner point of the set $\text{comp}(\mathfrak{g})$ of elliptic elements in $\mathfrak{g}$. This allows us to obtain a complete characterization of Lie algebras $\mathfrak{g}$, representations $\rho$ with discrete kernel and generators $X$ such that $\text{tr}(e^{i\partial\rho(X)})<\infty$.