Central decomposition of invariant states applications to the groups of time translations and of euclidean transformations in algebraic field theory |
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Authors: | D Kastler M Mebkhout G Loupias L Michel |
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Institution: | (1) UER Pluridisciplinaire de Marseille-Luminy, Université d'Aix-Marseille II et Centre de Physique Théorique, CNRS, Marseille, France;(2) Physique Mathematique Equipe de Recherche Associee au CNRS, Département de Mathématiques, Université des Sciences et Technique du Languedoc, Montpellier, France;(3) Institut des Hautes Etudes Scientifique, 91440 Bures-Sur-Yvette, France |
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Abstract: | With
aC*-algebra with unit andg G![rarr](/content/vh8245315314820q/xxlarge8594.gif)
g
a homomorphic map of a groupG into the automorphism group ofG, the central measure
of a state of
is invariant under the action ofG (in the state space of
) iff is -invariant. Furthermore if the pair {
,G} is asymptotically abelian, is ergodic iff
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
, the associated covariant representations of {
, } 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. |
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Keywords: | |
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