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

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

Pseudo-multiplicative unitaries associated with inclusions of finite-dimensional C*-algebras

AbstractWe construct a pseudo-multiplicative unitary from an inclusion of finite-dimensional C*-algebras when the inclusion satisfies certain conditions. We also study an example which provides a non-trivial pseudo-multiplicative unitary

C*-BUNDLES ASSOCIATED WITH GENERALIZED BRATTELI DIAGRAMS

We study the inductive limit C*-algebras associated with generalized Bratteli diagrams. For this purpose we construct sequences of locally trivial C*-bundles associated with their diagrams and study the inclusions of C*-bundles. We give a complete condition to ensure locally trivial inclusions. Then we study generalized Bratteli diagrams arising from sequences of covering maps. We show that their inductive limit C*-algebras come from lower semi-continuous cross sections of C*-bundles. Using this, we prove that they are simple if the sequences of covering maps are minimal. </jats:p