THE KMS CONDITION AND SPECTRAL PASSIVITY IN GROUP DUALITY

Let (A, G, α) be a C*-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in G ̂ and a finite constant κ such that Πi = 1n φ(xi*xi) ≤ κ Πi = 1n φ(xixi*) whenever xi lies in a spectral subspace Rα(Ωi), where Ω1 + ... + Ωn ⊂ Ω....