A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS

Source abstract

Recently, Neufang, Ruan and Spronk proved a completely isometric representation theorem for the measure algebra M(G) and for the completely bounded (Herz–Schur) multiplier algebra M cb A(G) on [Formula: see text], where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups 𝔾 = (M, Γ, φ, ψ). More precisely, we introduce the algebra [Formula: see text] of completely bounded right multipliers on L 1 (𝔾) and we show that [Formula: see text] can be identified with the algebra of normal completely bounded [Formula: see text]-bimodule maps on [Formula: see text] which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L 1 (𝔾) is in fact implemented by an element of [Formula: see text]. We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutation relation.