The KMS condition and spectral passivity in group duality (Q799908)

Let (A,G,\(\alpha)\) be a \(C^*\)-dynamical system, where G is abelian, and let \(\phi\) be an invariant state. Suppose that there is a neighbourhood \(\Omega\) of the identity in \(\hat G\) and a finite constant \(\kappa\) such that \(\prod^{n}_{i=1}\phi (x^*_ ix_ i)\leq\kappa \prod^{n}_{i=1}\phi (x_ ix^*_ i)\) whenever \(x_ i\) lies in a spectral subspace \(R^{\alpha}(\Omega_ i),\) where \(\Omega_ 1+\Omega_ 2+...+\Omega_ n\subseteq\Omega.\) This condition of complete spectral passivity, together with self-adjointness of the left kernel of \(\phi\), ensures that \(\phi\) satisfies the KMS condition for some one- parameter subgroup of G. This extends some concepts considered in the case \(G={\mathbb{R}}\) by \textit{J. de Cannière} [Commun. Math. Phys. 84, 187- 206 (1982; Zbl 0507.46053)] and the author [J. Funct. Anal. 46, 246-257 (1982; Zbl 0485.46032)].