Measured quantum groupoids associated with matched pairs of locally compact groupoids

Abstract

Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids. Contents 1. Introduction 2. Measured quantum groupoids and their actions 2.1. Measured quantum groupoids 2.2. Measured quantum groupoids in action 2.3. The abelian case 3. A generalization of the matched pair procedure 3.1. Measured matched pair of groupoids 3.2. Families of examples 3.3. The case of principal and transitive groupoids 3.4. The mutual actions of a matched pair of groupoids 3.5. A pseudo multiplicative unitary associated with a matched pair 4. The quantum groupoid structures associated with a matched pair 4.1. The coproduct 4.2. The co-involution 4.3. The Haar operator valued weights 5. Two families of examples 5.1. A matched pair of groups action on a space 5.2. The case of a principal and transitive groupoid References

Measured quantum groupoids associated to proper dynamical quantum groups

“…This Hilbert space can naturally be identified with the separated completions of the algebraic tensor products DpH β ,μq b H and H b DpH α ,μq with respect to the sesquilinear forms given by xξ b η|ξ 1 b η 1 y " xη|αpxξ|ξ 1 y β,μ qη 1 y and xξ b η|ξ 1 b η 1 y " xξ|βpxη|η 1 y α,μ qξ 1 y, (27) respectively, via…”

“…Proof. The maps Λ, Λ 1 are surjective because Λ ν pAq Ď H is dense, and they are well-defined and isometric because (27), (25) and (30) imply for all x, y P A xΛpx b yq|Λpx 1 b y 1 qy " νpx ˚spφpy ˚y1 qqx 1 q, xΛ 1 px b yq|Λ 1 px 1 b y 1 qy " νpx ˚rpB y pφpy ˚y1 qqqx 1 q.…”

“…Measured quantum groupoids were introduced by Enock, Lesieur and Vallin [2,12] to capture generalized Galois symmetries of certain inclusions of type II 1 factors [4,3,14]. Apart from this fundamental example in von Neumann algebra theory, which was also considered in the algebraic setting [8,18], and from the finite case, only few measured quantum groupoids have been constructed and investigated yet [12,27].…”

Morita equivalence of measured quantum groupoids. Application to deformation of measured quantum groupoids by 2-cocycles

“…The action a can be identified with an actionã of G 1 on L ∞ (X ×G 2 ) ([Val6], 5.1.2) and the crossed product L ∞ (G 2 ) ⋊ a G(G 1 ) can be identified with the crossed-product L ∞ (X × G 2 ) ⋊ã G 1 , which will be considered as bounded operators on L 2 (X × G × G) ([Val6], 5.1.1). We can identify L 2 (X × G 2 ) s 2 ⊗ r 1 L ∞ (X) L ∞ (X × G 1 ) with L 2 (X × G 2 ) ⊗ L 2 (G 1 ) ( [Val6], 5.1.1); using these identifications, are given in ([Val6] 5.1.2) the formulae of the coproduct Γ we can put on this crossed-product. For any f ∈ L ∞ (X × G 2 ), h ∈ L ∞ (X), k ∈ L ∞ (G 1 ), we have :…”

“…Measured quantum groupoids associated to a matched pair of groupoids. In [Val6] was decribed a procedure for constructing measured quantum groupoids : Let G be a locally compact groupoid, with G (0) as set of units, and r : G → G (0) (resp. s : G → G (0) ) as range (resp.…”