Closed quantum subgroups of locally compact quantum groups

Abstract

We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions -one due to S. Vaes and one due to S.L. Woronowicz -are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podleś from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.

Group-like projections for locally compact quantum groups

“…Let H and G be locally compact quantum groups. A morphism π ∈ Mor(C u 0 (G), C u 0 (H)) such that (π ⊗ π) • ∆ u G = ∆ u H • π is said to define a homomorphism from H to G. If π(C u 0 (G)) = C u 0 (H), then H is called Woronowicz -closed quantum subgroup of G [2]. A homomorphism from H to G admits the dual homomorphism π ∈ Mor(C u 0 ( H),…”

The Haagerup property for locally compact quantum groups

“…There are various notions of a "closed quantum subgroup" H of a locally compact quantum group G in the literature and these are analysed in detail in the recent article [24]. The weakest of these is when maps surjectively onto C 0 .H/, in which case H is called a closed quantum subgroup of G in the sense of Woronowicz.…”

Lattice of Idempotent States on a Locally Compact Quantum Group

“…, it is the right regular representation in the terminology of [20]) and W P MpC 0 p p Gq b C u 0 pGqq is its "half-lifted" version (cf. [19,4]).…”