Convolution of Trace Class Operators over Locally Compact Quantum Groups

We study locally compact quantum groups G through the convolution algebras L-1(G) and (T(L-2(G)), (sic)). We prove that the reduced quantum group C*-algebra C-0(G) can be recovered from the convolution (sic) by showing that the right T(L-2(G))-module < K(L-2(G)) (sic) T(L-2(G))> is equal to C-0(G). On the other hand, we show that the left T(L-2(G))-module < T(L-2(G)) (sic) K(L-2(G))> is isomorphic to the reduced crossed product C-0((G) over cap)(r) proportional to C-0(G), and hence is a much larger C*-subalgebra of B(L-2(G)). We establish a natural isomorphism between the completely bounded right multiplier algebras of L-1(C) and (T(L-2(G)), (sic)), and settle two invariance problems associated with the representation theorem of Junge-Neufang-Ruan (2009). We characterize regularity and discreteness of the quantum group Gin terms of continuity properties of the convolution (sic) on T(L-2(G)). We prove that if G is semi-regular, then the space < T(L-2(G)) (sic) B(L-2(G))> of right G-continuous operators on L-2(G), which was introduced by Bekka (1990) for L-infinity (G), is a unital C*-subalgebra of B(L-2(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(G)). We also characterize some commutation relations of completely bounded multipliers of (T(L-2(G)), (sic)) over B(L-2(G)).