On a two-sided version of Reiter's condition and weak containment

1978 ◽  
Vol 31 (1) ◽  
pp. 605-610 ◽  
Author(s):  
H. L. Skudlarek
Keyword(s):  
Weak Containment

scholarly journals Dynamical properties of profinite actions

2011 ◽  
Vol 32 (6) ◽  
pp. 1805-1835 ◽  
Author(s):  
MIKLÓS ABÉRT ◽  
GÁBOR ELEK
Keyword(s):  
Finite Index ◽  
Linear Groups ◽  
Regular Graphs ◽  
Weak Containment ◽  
Residually Finite Groups ◽  
The Family ◽  
Property T ◽  
Free Actions ◽  
Theoretical Consequence ◽  
Lattice Of Subgroups

AbstractWe study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

Amenability and Co-Amenability for Locally Compact Quantum Groups

2003 ◽  
Vol 14 (08) ◽  
pp. 865-884 ◽  
Author(s):  
E. Bédos ◽  
L. Tuset
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Weak Containment ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqg's.

scholarly journals Induction for locally compact quantum groups revisited

2019 ◽  
Vol 150 (2) ◽  
pp. 1071-1093
Author(s):  
Mehrdad Kalantar ◽  
Paweł Kasprzak ◽  
Adam Skalski ◽  
Piotr M. Sołtan
Keyword(s):  
Quantum Groups ◽  
General Setting ◽  
Locally Compact ◽  
Induced Representations ◽  
Open Quantum ◽  
Locally Compact Quantum Groups ◽  
Weak Containment ◽  
Compact Quantum Groups

AbstractIn this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

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