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Actions, quotients and lattices of locally compact quantum groups. (English) Zbl 1466.46064

Property (T) for a locally compact group says that any unitary representation that has almost invariant vectors also has invariant vectors. This property also makes sense for locally compact quantum groups. This article proves several permanence results for property (T) for general locally compact quantum groups that are well known for groups. Some of them have already been proven for discrete quantum groups. This is used to produce some new examples of locally compact quantum groups with property (T). Along the way, the article also develops tools to study the representation theory of locally compact quantum groups, which may also be useful for other purposes than the study of property (T).
An important result about property (T) for groups says that if \(H\) is a subgroup in \(G\) of finite covolume, then \(H\) has property (T) if and only if \(G\) has it. This equivalence is extended here to locally compact quantum groups. As a preparation, one must define when a quantum subgroup has finite covolume. And this, in turn, requires the canonical measure on the homogeneous space \(G/H\), which exists if the modular function of \(G\) restricted to \(H\) is the modular function of \(H\). This article carries this result over to locally compact quantum groups. Then “finite covolume” means that the resulting unique invariant weight is finite. If such a finite invariant weight exists, then it is shown that the quantum subgroup is unimodular if and only if the ambient quantum group is unimodular.
Property (T) also has a relative version for a pair of locally compact (quantum) groups, saying that a representation with almost invariant vectors for the larger quantum group has true invariant vectors for the smaller quantum group. It is shown that this relative version for a normal quantum subgroup is equivalent to property (T) for the quotient quantum group.
The Drinfeld double construction puts together a quantum group and its dual in a canonical way. A discrete quantum group is a quantum subgroup of finite covolume in its Drinfeld double if and only if it is of Kac type. Since discrete quantum groups with property (T) must be of Kac type by a result of P. Fima [Int. J. Math. 21, No. 1, 47–65 (2010; Zbl 1195.46072)], it follows that the Drinfeld double of a discrete quantum group with property (T) inherits property (T).

MSC:

46L67 Quantum groups (operator algebraic aspects)
20G42 Quantum groups (quantized function algebras) and their representations
46L65 Quantizations, deformations for selfadjoint operator algebras
46L30 States of selfadjoint operator algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Citations:

Zbl 1195.46072
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