this entails that $M_{01} \Omega = 0$ (and hence, by Lemma \[invariance\], $\Omega$ is invariant under the entire group $U({\mbox{SO$_0(2,n\!-\!1)$}})$), and $\omega$ is either [@PuWo Theorem 1.3] a ground state for $M_{01}$ (which is excluded by Lemma \[nogroundstate\]), or satisfies, for some [*a priori*]{} unknown $\beta \geq 0$, the Kubo–Martin–Schwinger (KMS) condition. In fact, our assumptions exclude the possibility of $\beta = 0$. In the proof of Lemma 4.1 in [@PuWo] it is shown that if $\beta = 0$, then either $\omega$ is a trace state on $\As(W_R)$ or $M_{01} = 0$. In the second case, one would have the triviality of the representation of the boost group and thus the triviality of $U({\mbox{SO$_0(2,n\!-\!1)$}})$, which is excluded by (i). The first case is excluded by assumption (iv). Therefore, for any pair of operators $A,B \in \As(W_R)$ there exists an analytic function $F$ in the strip $\{ z \in \CC \mid 0 < \Im(z) < \beta \}$ with continuous boundary values at $\Im(z) = 0$ and $\Im(z) = \beta$, which are given by $$\label{kms}
F(t) = \omega( AB(t)) \, , \quad
F(t+i\beta) = \omega( B(t)A) \, ,$$ respectively, for all $t \in \RR$ and with $B(t) \doteq e^{itM_{01}}Be^{-itM_{01}}$. By the –covariance 