Convolution of Trace Class Operators over Locally Compact Quantum Groups

Hu, Zhiguo ; Neufang, Matthias ; Ruan, Zhong-Jin

Canadian journal of mathematics, Tome 65 (2013) no. 5, pp. 1043-1072

Résumé

We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\triangleright\right)$ . We prove that the reduced quantum group ${{C}^{*}}$ -algebra ${{C}_{0}}\left( \mathbb{G} \right)$ can be recovered from the convolution $\triangleright $ by showing that the right $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ -module $\left\langle K\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,T\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is equal to ${{C}_{0}}\left( \mathbb{G} \right)$ . On the other hand, we show that the left $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ -module $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,K\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is isomorphic to the reduced crossed product ${{C}_{0}}\left( \widehat{\mathbb{G}} \right){{\,}_{r}}\,\ltimes \,{{C}_{0}}\left( \mathbb{G} \right)$ , and hence is a much larger ${{C}^{*}}$ -subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ .We establish a natural isomorphism between the completely bounded right multiplier algebras of ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright\right)$ , and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright $ on $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ . We prove that if $\mathbb{G}$ is semiregular, then the space $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,B\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ of right $\mathbb{G}$ -continuous operators on ${{L}_{2}}\left( \mathbb{G} \right)$ , which was introduced by Bekka (1990) for ${{L}_{\infty }}\left( G \right)$ , is a unital ${{C}^{*}}$ -subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ . In the representation framework formulated by Neufang–Ruan–Spronk (2008) and Junge–Neufang–Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ . We also characterize some commutation relations of completely bounded multipliers of $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright\right)$ over $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$ .

Détail

DOI : 10.4153/CJM-2012-030-5

Mots-clés : 22D15, 43A30, 46H05, locally compact quantum groups and associated Banach algebras