Quantum groupoids of compact type. (English) Zbl 1071.46041

Summary: To any groupoid, equipped with a Haar system, J. M. Vallin [J. Oper. Theory 35, No. 1, 39–65 (1996; Zbl 0849.22002) and ibid. 44, No. 2, 347–368 (2000; Zbl 0986.22002)] has associated several objects (pseudo-multiplicative unitary, Hopf-bimodule) in order to generalize, up to the groupoid case, the classical notions of multiplicative unitary and Hopf-von Neumann algebra, which were intensely used to construct quantum groups in the operator algebra setting. In two former articles [M. Enock, J. Funct. Anal. 178, No. 1, 156–225 (2000; Zbl 0982.46046) and M. Enock and J. M. Vallin, ibid. 172, No. 2, 249–300 (2000; Zbl 0974.46055)], starting from a depth-2 inclusion of von Neumann algebras, we have constructed such objects, which allowed us to study two ‘quantum groupoids’ dual to each other. We are now investigating in greater details the notion of pseudo-multiplicative unitary, following the general strategy developed by Baaj and Skandalis for multiplicative unitaries.


46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22A22 Topological groupoids (including differentiable and Lie groupoids)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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