Arens irregularity of the trace class convolution algebra

For any locally compact quantum group G, the space T(L-2(G)) of trace class operators on L-2(G) is a Banach algebra with the convolution induced by the right fundamental unitary of G. We study certain Arens irregularity properties of this convolution algebra. It is shown in particular that T(L-2(G)) is right strongly Arens irregular in the sense of Dales and Lau (The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005)) if and only if G is finite. This generalizes a result by the second author on locally compact groups. We obtain a natural class of Banach algebras for which Arens regularity and strong Arens irregularity are, surprisingly, equivalent. We also give a precise description of the right topological centre of T(L-2(G))** for all infinite discrete quantum groups G with L-1(G) strongly Arens irregular.