On the similarity problem for locally compact quantum groups |
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Authors: | Michael Brannan Sang-Gyun Youn |
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Institution: | 1. Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA;2. Department of Mathematical Sciences, Seoul National University, Gwanak-Ro1, Gwanak-Gu, Seoul 08826, Republic of Korea |
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Abstract: | A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ?-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type. |
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Keywords: | 20G42 22D12 22D15 46L07 46L89 81R50 Locally compact quantum group Amenability Completely bounded representation Day–Dixmier property |
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