Central decomposition of invariant states applications to the groups of time translations and of euclidean transformations in algebraic field theory

Abstract

With$\mathfrak{A}$ aC*-algebra with unit andg∈G→α _g a homomorphic map of a groupG into the automorphism group ofG, the central measureμ _Φ of a state Φ of$\mathfrak{A}$ is invariant under the action ofG (in the state space of$\mathfrak{A}$) iff Φ is α-invariant. Furthermore if the pair {$\mathfrak{A}$,G} is asymptotically abelian, Φ is ergodic iffμ _Φ is ergodic. Transitive ergodic states (corresponding to transitive central measures) are centrally decomposed into primary states whose isotropy groups form a conjugacy class of subgroups. IfG is locally compact and acts continuously on$\mathfrak{A}$, the associated covariant representations of {$\mathfrak{A}$, α} are those induced by such subgroups. Transitive states under time-translations must be primary if required to be stable. The last section offers a complete classification of the isotropy groups of the primary states occurring in the central decomposition of euclidean transitive ergodic invariant states.