C*-pseudo-multiplicative unitaries, Hopf C*-bimodules and their Fourier algebras

Abstract

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-bimodules studied by Enock, Lesieur and Vallin. To each C * -pseudomultiplicative unitary, we associate two Fourier algebras with a duality pairing and in the regular case two 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.

Measured quantum groupoids associated to proper dynamical quantum groups

“…To show that ν admits a bounded GNS-representation and to lift the comultiplication to the level of operator algebras, we use a fundamental unitary. To take full advantage of this unitary, we describe its domain and range as relative tensor products, and show that it is a pseudomultiplicative unitary in the sense of [23] and [25]. The necessary modules are introduced in §2.2, and the unitary itself is constructed in §2.3.…”

“…To show that it is a C˚pseudo-multiplicative unitary in the sense of [23], we only need to prove: 2.6.1 Proposition. The following equations for subspaces of LpH,H E :…”

“…To obtain a C ˚-algebra or von Neumann algebra, one has to show that these multiplication operators are bounded. To prove this and to lift the comultiplication to the resulting C ˚-algebra and von Neumann algebra, we proceed as in the case of quantum groups [22] and construct a fundamental unitary which is pseudo-multiplicative on the level of von Neumann algebras and C ˚-algebras in the sense of [25] and [23], respectively. The general theory of these unitaries then yields completions of the dynamical quantum group in the form a Hopf C ˚-bimodule and a Hopf von-Neumann bimodule, and simultaneously a Pontrjagin dual in the same form.…”

“…Theorem. W and V are C˚-pseudo-multiplicative unitaries in the sense of[23].Proof. The assertion on W is Proposition 2.6.1 and Lemma 2.5.2.…”

Integration on algebraic quantum groupoids

“…In the involutive case, we can construct operator-algebraic completions in the form of Hopf-von Neumann bimodules [20] and of Hopf C * -bimodules [15], and thus link the algebraic approaches to quantum groupoids to the operator-algebraic one. This construction generalizes corresponding results of Kustermans and Van Daele for algebraic quantum groups [9] and of the author for dynamical quantum groups [17], and is detailed in a forthcoming article [13].…”