Coideals, Quantum Subgroups and Idempotent States

Abstract

We establish a one to one correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra L ∞ (G) that are preserved by the scaling group. In particular we show that there is a one to one correspondence between idempotent states on G and ψ G -expected left-invariant von Neumann subalgebras of L ∞ (G). We characterize idempotent states of Haar type as those corresponding to integrable normal coideals preserved by the scaling group. We also establish a one to one correspondence between open subgroups of G and central idempotent states on the dual G. Finally we characterize coideals corresponding to open quantum subgroups of G as those that are normal and admit an atom. As a byproduct of this study we get a number of universal lifting results for Podleś condition, normality and regularity and we generalize a number of results known before to hold under the coamenability assumption.2010 Mathematics Subject Classification. Primary: 46L65 Secondary: 43A05, 46L30, 60B15.

Group-like projections for locally compact quantum groups

“…The next theorem was first proved in [4,Theorem 5.15]. The previous proof strongly uses the universal C * -version of G. In what follows we give a simpler proof which is based on the von Neumann version of G. Proof.…”

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

“…Suppose that N is preserved by τ . Using [11,Theorem 4.2] and [2, Theorem 3.1] we conclude that P is τ -invariant (a direct proof can also be obtained as in the proof of Corollary 5.5).…”

“…Ad(2): Let y ∈ L ∞ ( G) ′ be an element for which there exist µ ∈ L ∞ (G) * , such that yη(z) = η((µ ⊗ id)(∆(z)) for all z ∈ D(η) (the set of such y's is dense in L ∞ ( G) ′ , see e.g. [11,Equation 4.4]). For all x ∈ X ∩ D(η) we have yη(x) = η((µ ⊗ id)(∆(x)) ∈ L 2 (X), i.e.…”

“…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.…”

“…In order to relate N with ω and P we shall also use notation N ω , N P etc. The following relations are scattered throughout [2] and [11]…”