Group-like projections for locally compact quantum groups

Abstract

Let G be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in L ∞ (G) preserved by the scaling group and idempotent states on the dual quantum group G. As a byproduct we give a simple proof that normal integrable coideals in L ∞ (G) which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of G.

Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

“…Moving on to the realm of locally compact quantum groups, Faal and the author of the present paper extended the first half of Illie-Spronk result proving a 1-1 correspondence between group-like projections on L ∞ (G) preserved by the scaling group, and idempotent states on the dual locally compact quantum group G, see [2]. One of the main result of the present paper provides the quantum group extension of the second half the Illie-Spronk result by giving characterization of contractive idempotent functionals on G in terms of shifts of group-like projections preserved by the scaling group.…”

“…Idempotent states on quantum groups has been intensely studied by now, see e.g. [4], [21], [20], [11], [2]. Note that if µ ∈ C u 0 (G) * , µ = 0 (here µ is a functional, not necessarily a state) satisfies µ * µ = µ and µ ≤ 1 then µ = 1.…”

“…where ∆ (2) = (∆ ⊗ id) • ∆. Using [2, Theorem 2.3] (where the group G × G acts on L ∞ (G) by shifts from the left and the right) we obtain x ∈ {ω * x * µ : ω, µ ∈ L ∞ (G) * } ′′ and thus using (0.7) we get π(x) = ρ(x).…”

Lattice of Idempotent States on a Locally Compact Quantum Group

“…By Theorem 4.11 and the fact that Q K ω is σ ψ -invariant (Remark 4.7) we have It is easy to see that r ω is idempotent (P ω is a group-like projection, see the proof of Theorem 4.11). Additionally, by [6,Theorem 3…”

“…Idempotent states on locally compact quantum groups have been investigated in a number of papers, e.g. [25,8,7,2,28,9,27,29,13,6]. The main idea behind these investigations was based on the classical result of Kawada and Itô which establishes a bijection between idempotent states on a classical locally compact group G and compact subgroups of G with the state given by integration with respect to the Haar measure of the subgroup.…”