Abstract
LetF be a closed face of the weak* compact convex state space of a unitalC*-algebraA. The class ofF-abelian states, introduced earlier by the author, is studied further. It is shown (without any restriction onA orF) thatF is a Choquet simplex if and only if every state inF isF-abelian, and that it is sufficient for this that every pure state inF isF-abelian. As a corollary, it is deduced that an arbitraryC*-dynamical system (A, G, α) isG-abelian if and only if every ergodic state is weakly clustering. Nevertheless the set of allF-abelian (or evenG-abelian) states is not necessarily weak* compact.
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Communicated by H. Araki
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Batty, C.J.K. Abelian faces of state spaces ofC*-algebras. Commun.Math. Phys. 75, 43–50 (1980). https://doi.org/10.1007/BF01962590
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DOI: https://doi.org/10.1007/BF01962590