TY - JOUR
T1 - A representation theorem for locally compact quantum groups
AU - Junge, Marius
AU - Neufang, Matthias
AU - Ruan, Zhong Jin
N1 - Funding Information:
The first and third authors were partially supported by the National Science Foundation (NSF) DMS 05-56120 and DMS 05-00535. The second author was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) 261894-2003.
PY - 2009/3
Y1 - 2009/3
N2 - 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.
AB - 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.
KW - Completely bounded module maps
KW - Completely bounded multiplier algebra
KW - Completely isometric representation
KW - Locally compact quantum group
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U2 - 10.1142/S0129167X09005285
DO - 10.1142/S0129167X09005285
M3 - Article
AN - SCOPUS:65349120824
SN - 0129-167X
VL - 20
SP - 377
EP - 400
JO - International Journal of Mathematics
JF - International Journal of Mathematics
IS - 3
ER -