For any locally compact quantum group G, the space T(L2(G)) of trace class operators on L2(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(L2(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(L2(G))** for all infinite discrete quantum groups G with L1(G) strongly Arens irregular.

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Journal Bulletin of the London Mathematical Society
Hu, Z. (Zhiguo), Neufang, T, & Ruan, Z.-J. (Zhong-Jin). (2013). Arens irregularity of the trace class convolution algebra. Bulletin of the London Mathematical Society, 45(2), 351–362. doi:10.1112/blms/bds093