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Multiplicative partial isometries and finite quantum groupoids. (English) Zbl 1171.47306
Vainerman, Leonid (ed.), Locally compact quantum groups and groupoids. Proceedings of the 69th meeting of theoretical physicists and mathematicians, Strasbourg, France, February 21–23, 2002. Berlin: Walter de Gruyter (ISBN 3-11-017690-4/pbk). IRMA Lect. Math. Theor. Phys. 2, 189-227 (2003).
Summary: We continue, after J.–M. Vallin [J. Algebra 239, No. 1, 215–161 (2001; Zbl 1003.46040)], the study of multiplicative partial isometries over a finite-dimensional Hilbert space. We prove that, after an ampliation and a reduction, any regular multiplicative partial isometry is isomorphic to an irreducible one. For this irreducible multiplicative partial isometry, we prove quantum Markov properties. Namely, both normalized Haar measures of the quantum groupoids associated to a multiplicative partial isometry can be extended to a unique faithful positive linear form on the involutive algebra generated by these groupoids (the Weyl algebra). Using this Markov extension, a multiplicative partial isometry can be expressed as a composition of two very simple partial isometries. The two Haar conditional expectations of the quantum groupoids with values in the intersection of their algebras can be, in a unique way, extended to a multiplicative conditional expectation on the Weyl algebra; moreover, this extension is invariant with respect to the Markov extension of the Haar measures. We prove that a multiplicative partial isometry is completely determined by the two quantum groupoids in duality which it generates and the spaces of fixed and cofixed vectors. Finally, we give a complete characterization of quantum groupoids in duality acting on the same Hilbert space in the irreducible situation.
For the entire collection see [Zbl 1005.00029].

##### MSC:
 47L40 Limit algebras, subalgebras of $$C^*$$-algebras 46L05 General theory of $$C^*$$-algebras 46L37 Subfactors and their classification 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 46L65 Quantizations, deformations for selfadjoint operator algebras 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory