A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS
A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS
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DOI：
10.1142/s0129167x09005285
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发表时间：
2009-03
影响因子：
0.6
通讯作者：
M. Junge;M. Neufang;Z. Ruan
中科院分区：
文献类型：
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作者：
M. Junge;M. Neufang;Z. Ruan
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...
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...