Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families I

Abstract

Abstract. Pseudo-multiplicative unitaries on C*-modules generalize the multiplicative unitaries of Baaj and Skandalis [1], and are analogues of the pseudo-multiplicative unitaries on Hilbert spaces studied by Enock, Lesieur, Vallin [5], [10], [21]. We introduce Hopf C*-families on C*-bimodules and associate to special classes of pseudo-multiplicative unitaries two Hopf C*-families. Furthermore, we discuss dual pairings and counits on these Hopf C*-families, étalé and proper pseudo-multiplicative unitaries, and two classes of examples. In a later article, we will study regularity conditions on pseudo-multiplicative unitaries, coactions of Hopf C*-families on C*-algebras, and duality.Mathematics Subject Classification (2000). 46L55; 46L08, 22A22, 46L05.

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

“…The theory presented here overcomes several restrictions of our former generalizations of multiplicative unitaries and Hopf C * -algebras [27] (see also [26]). It was applied already in [31] to the definition and study of compact C * -quantum groupoids, and in [30] to the study of reduced crossed products for coactions of Hopf C * -bimodules on C * -algebras and to an extension of the Baaj-Skandalis duality theorem.…”

“…The theory presented here overcomes several restrictions of our former generalizations of multiplicative unitaries and Hopf C * -algebras [34]; see also [33].…”

“…To every locally compact, Hausdorff, second countable groupoid with left Haar system, we shall associate a C * -pseudo-multiplicative unitary. The underlying pseudomultiplicative unitary was introduced by Vallin [38], and associated unitaries on C *modules were discussed in [23,34]. We focus on the aspects that are new in the present setting.…”

“…In the setting of von Neumann algebras, the targets of the comultiplications can be described using Sauvageot's fiber product [28,37]. For the setting of C * -algebras, a partial solution was proposed in [34], and a general fiber product construction which suits our purpose was introduced in [31].…”

“…Let G be a locally compact, Hausdorff, second countable groupoid with left Haar system λ and associated right Haar system λ −1 , and let µ be a measure on G 0 with full support. We associate to this data a C * -pseudo-multiplicative unitary such that the underlying pseudo-multiplicative unitary and the associated unitary on C * -modules are the ones introduced by Vallin [33] and O'uchi [19,27], respectively. We focus on the aspects that are new in the present setting.…”