• DOI:10.1142/S0129167X09005285
• Corpus ID: 53399978

A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS

@article{Junge2009ART,
  title={A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS},
  author={Marius Junge and Matthias Neufang and Zhong‐Jin Ruan},
  journal={International Journal of Mathematics},
  year={2009},
  volume={20},
  pages={377-400}
}

• M. Junge, M. Neufang, Z. Ruan
• Published 1 March 2009
• Mathematics
• International Journal of Mathematics

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 $\mathcal{B}(L_{2}(G))$, where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups 𝔾 = (M, Γ, φ, ψ). More precisely, we introduce the algebra $M_{\rm cb}^{r} (L_1(\mathbb{G}))$ of completely bounded right multipliers on L1(𝔾) and we show that $M^r_{\rm cb} (L_1(\mathbb{G}))$ can be identified with the algebra of normal completely bounded $\hat{M}$-bimodule maps on $\mathcal{B}(L_2(\mathbb{G}))$ which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L1(𝔾) is in fact implemented by an element of $M_{\rm cb}^r (L_1(\mathbb{G}))$. We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutatio...