A representation theorem for locally compact quantum groups

Marius Junge, Matthias Neufang, Zhong Jin Ruan

Research output: Contribution to journalArticlepeer-review


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 McbA(G) on B(L 2(G)), where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups double-strock G sign = (M, Γ, φ, ψ). More precisely, we introduce the algebra Mcbr(L1(G)) of completely bounded right multipliers on L1(double-strock G sign) and we show that $Mcbr(LL(G)) can be identified with the algebra of normal completely bounded M̂-bimodule maps on B(L2(G)) which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L1(double-strock G sign) is in fact implemented by an element of Mcbr(L1(G)). We also show that our representation framework allows us to express quantum group " Pontryagin" duality purely as a commutation relation.

Original languageEnglish (US)
Pages (from-to)377-400
Number of pages24
JournalInternational Journal of Mathematics
Issue number3
StatePublished - Mar 2009


  • Completely bounded module maps
  • Completely bounded multiplier algebra
  • Completely isometric representation
  • Locally compact quantum group

ASJC Scopus subject areas

  • General Mathematics


Dive into the research topics of 'A representation theorem for locally compact quantum groups'. Together they form a unique fingerprint.

Cite this