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Article Dans Une Revue Journal of Noncommutative Geometry Année : 2019

The Baum–Connes property for a quantum (semi-)direct product

Résumé

The well known "associativity property" of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete quantum groups. This decomposition allows to define an appropriate triangulated functor relating the Baum-Connes property for the quantum semi-direct product to the Baum-Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum-Connes property generalizes the result [5] of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. The K-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena. The analogous strategy can be applied for the dual of a quantum direct product. In this case, we obtain, in addition, a connection with the Künneth formula, which is the quantum counterpart to the result [7] of J. Chabert, S. Echterhoff and H. Oyono-Oyono. Again the K-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena.
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Dates et versions

hal-04033081 , version 1 (16-03-2023)

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Rubén Martos. The Baum–Connes property for a quantum (semi-)direct product. Journal of Noncommutative Geometry, 2019, 13 (4), pp.1295-1357. ⟨10.4171/jncg/348⟩. ⟨hal-04033081⟩
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