Around Property (T) for quantum groups

Abstract

We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals Property (T) is shown to be equivalent to Property (T)${}^{1,1}$ of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on 'typical' representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image.

53 pages; v2: made several structural changes and added more material, in particular Propositions 2.14, 2.15 and 3.5 and Corollary 3.6, solving affirmatively the question raised in former Remark 1.29; to appear in Communications in Mathematical Physics

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Cited by in corpus (9)

• Operator algebras in rigid C*-tensor categories
• Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms
• Admissibility Conjecture and Kazhdan's Property (T) for quantum groups
• The Fourier algebra of a rigid $C^{\ast }$-tensor category
• Quantum groups, property (T), and weak mixing
• Howe-Moore type theorems for quantum groups and rigid C*-tensor categories
• Compact quantum groups with representations of bounded degree
• Generating functionals for locally compact quantum groups
• Property (T), property (F) and residual finiteness for discrete quantum groups