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Systèmes dynamiques non commutatifs et moyennabilité. (Non commutative dynamical systems and amenability). (French) Zbl 0608.46036
The notion of positive type function with respect to a \(C^ *\)-dynamical system (A,G,\(\alpha)\) is introduced and studied. When A is a von Neumann algebra and G is a discrete group, a new characterization of amenability for the \(W^ *\)-dynamical systems (A,G,\(\alpha)\) is given in terms of positive type functions of this kind. Amenability is also characterized by topological properties of the set of unitary cocycles defined on the Borel G-space which is associated to the G-action on the centre of A. Assuming that A is a \(C^ *\)-algebra, we study a condition which implies the equality of the reduced crossed product relative to the \(C^ *\)- dynamical system with the crossed product. When A is nuclear, this condition is equivalent to the nuclearity of the crossed products.

MSC:
46L55 Noncommutative dynamical systems
46L10 General theory of von Neumann algebras
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