C*-pseudo-multiplicative unitaries and Hopf C*-bimodules

Details C*-pseudo-multiplicative unitaries and Hopf C*-bimodules C*-pseudo-multiplicative unitaries and Hopf C*-bimodules

2009-08-13

arxiv.org/abs/0908.1850v1

Mathematics

Operator Algebras

50 pages; this is a substantial revision and expansion of the preprint "C*-pseudo-multiplicative unitaries" (arXiv:0709.2995)

Scientific paper

We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf--von Neumann-bimod-ules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing, a C*-tensor category of representations, and in the regular case two reduced and two universal Hopf C*-bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.

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