$C^*$-pseudo-Kac systems and duality for coactions of concrete Hopf $C^*$-bimodules

Abstract

We study coactions of concrete Hopf C ¦ -bimodules in the framework of (weak) C ¦pseudo-Kac systems, define reduced crossed products and dual coactions, and prove an analogue of Baaj-Skandalis duality.

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

“…A nondegenerate finite-dimensional concrete (shortly nfc.) C ¦ -B H B -algebra ÔH,A, αÕ consists of a finite-dimensional Hilbert space H, a nondegenerate C ¦ -algebra A LÔHÕ, and a [13,12]. We denote the set of such morphisms by MorÔAα, B β Õ. LÔπÔpiÕKÕ denotes the restriction of π.…”

“…In particular, we associate to every regular C * -pseudo-multiplicative unitary two Hopf C * -bimodules and two Fourier algebras with a duality pairing, and construct universal Hopf C * -bimodules from a C * -tensor category of representations of the unitary. The theory presented here was applied already in [30] to the definition and study of compact C * -quantum groupoids, and will be applied in a forthcoming article 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; see also [32].…”

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

“…But the theory developed there applies only to a special class of quantum groupoids that are analogues of r-discrete groupoids. Recently, we introduced a general definition of Hopf C * -bimodules and C * -pseudo-multiplicative unitaries [12,13,14] that, we hope, should provide the right basis for the study of quantum groupoids on the level of C * -algebras. The purpose of this article is to explain how the special theory developed in [11,15] fits into the general framework introduced in [13].…”

“…The approach in [11,15], however, is restricted to "decomposable" quantum groupoids which generalize r-discrete groupoids. Recently, we developed a general approach [12,13] that covers all locally compact groupoids. In this article, we explain how the special theory of [11,15] embeds into the general one of [12,13].…”

“…Recently, we developed a general approach [12,13] that covers all locally compact groupoids. In this article, we explain how the special theory of [11,15] embeds into the general one of [12,13].…”