Property T for general locally compact quantum groups

2015/09/08 by Chen, Xiao, Ng, Chi-Keung
#46L89 Secondary 22D25 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 20G42 #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1509.02262

Abstract

In this short article, we obtained some equivalent formulations of property T for a general locally compact quantum group \mathbbG, in terms of the full quantum group C^*-algebras C₀^u(\widehat\mathbbG) and the *-representation of C₀^u(\widehat\mathbbG) associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if \mathbbG is of Kac type, we show that \mathbbG has property T if and only if every finite dimensional irreducible *-representation of C₀^u(\widehat\mathbbG) is an isolated point in the spectrum of C₀^u(\widehat\mathbbG) (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property T discrete quantum groups using bicrossed products.

Related

• From Quantum Groups to Groups
• Quasi-Discrete Locally Compact Quantum Groups
• On certain invariants of compact quantum groups
• Locally compact quantum groups in the universal setting
• Inner amenability and approximation properties of locally compact quantum groups