Extremal invariant states with a spectrum condition and Borchers' theorem

Abstract

If the energy spectrum of an extremal invariant state ω is not the whole real line, it is shown that ω is either pure or uniquely decomposed into mutually disjoint pure states in the way that ω=λ-1F0λφοαtdt where φ is a pure state satisfying φοαλ=φ with λ>0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.