Expand all Collapse all | Results 1 - 1 of 1 |
1. CJM 2012 (vol 65 pp. 1043)
Convolution of Trace Class Operators over Locally Compact Quantum Groups We study locally compact quantum groups $\mathbb{G}$ through the
convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})),
\triangleright)$. We prove that the reduced quantum group
$C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution
$\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module
$\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is
equal to $C_0(\mathbb{G})$. On the other hand, we show that the left
$T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright
K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product
$C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is
a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$.
We establish a natural isomorphism between the completely bounded
right multiplier algebras of $L_1(\mathbb{G})$ and
$(T(L_2(\mathbb{G})), \triangleright)$, 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 $T(L_2(\mathbb{G}))$. We prove
that if $\mathbb{G}$ is semi-regular, then the space
$\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right
$\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was
introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra
of $B(L_2(\mathbb{G}))$. 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 $B(L_2(\mathbb{G}))$. We also characterize some commutation
relations of completely bounded multipliers of $(T(L_2(\mathbb{G})),
\triangleright)$ over $B(L_2(\mathbb{G}))$.
Keywords:locally compact quantum groups and associated Banach algebras Categories:22D15, 43A30, 46H05 |