Noncommutative Furstenberg boundary

Kalantar, Mehrdad; Kasprzak, Paweł; Skalski, Adam; Vergnioux, Roland

doi:10.48550/arxiv.2002.09657

Cited by 5 publications

(28 citation statements)

References 22 publications

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“…Their work led to a number of important results (see e.g. [9], [34], [12] and [48]), some of which we will make use of to pick up operator algebraic implications of our earlier results. One of the main results in [49] states that a discrete group G is C * -simple if and only if it admits a topologically free action on some G-boundary.…”

Section: 2mentioning

confidence: 97%

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Topological boundaries of connected graphs and Coxeter groups

Klisse¹

2020

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We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification [32]. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. In that context we speak of the compactification and the boundary of the Coxeter group. They also appear in [13] (see also [54]) and [53]. As it turns out, the canonical action of the group on its Cayley graph induces a natural action on the compactification and the boundary. We prove its amenability, we characterize when the compactification is small at infinity and we study classes of Coxeter groups for which the action is a topological boundary action in the sense of Furstenberg.The second part of the paper deals with the applications of our results to the study of (Iwahori) Hecke algebras. These are certain deformations of group algebras of Coxeter groups. We first study embeddings of Hecke C * -algebras and prove property Akemann-Ostrand for a certain class of Hecke-von Neumann algebras. Lastly, we make use of results that are widely related to Kalantar-Kennedy their approach to the C * -simplicity problem [49] to study the simplicity and injective envelopes of operator algebras associated with Hecke algebras.

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Section: 2mentioning

confidence: 97%

“…A series of breakthrough works followed (see e.g. [9], [34], [48] and also [5], [40], ...). Using Theorem 0.4 we will apply some of the results in the context of operator algebras associated with Hecke algebras.…”

Section: Introductionmentioning

confidence: 99%

Topological boundaries of connected graphs and Coxeter groups

Klisse¹

2020

Preprint

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“…For the moment we will consider quotients of closed quantum subgroups. The following notion was formulated in [23].…”

Section: Quantum Cosets Of Compact Quasi-subgroupsmentioning

confidence: 99%

On Ideals of $L^1$-algebras of Compact Quantum Groups

Benjamin¹

2021

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We develop a notion of a non-commutative hull for a left ideal of the L 1 -algebra of a compact quantum group G. A notion of non-commutative spectral synthesis for compact quantum groups is achieved as well. It is shown that a certain Ditkin's property at infinity (which includes those G where G has the approximation property) is equivalent to every hull having synthesis. We use this work to extend recent work of White that characterizes the weak * closed ideals of a measure algebra of a compact group to those of the measure algebra of a coamenable compact quantum group. In the sequel, we use this work to study bounded right approximate identities of certain left ideals of L 1 (G) in relation to coamenability of G.Acknowledgements: I thank Nico Spronk, Brian Forrest, and Michael Brannan for their supervision and support over the fruition of this project. I would like to thank Nico especially for his careful reading of this article and helpful comments. I would also like to thank Jared White for initiating the Groups, Operators, and Banach Algebras (GOBA) webinar in the Winter of 2020. The ideas that built this work were seeded from Jared's talk at the onset of the GOBA webinar.

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“…This connection had not attracted any attention until a few years ago, when it was rediscovered by Kalantar and Kennedy in their work on C * -simple groups [KK17]. Since then Hamana's construction of injective envelopes has been used to develop a Furstenberg boundarytype theory in several different contexts -for unitary representations of discrete groups [BK19], étale groupoids [Bor19], discrete quantum groups [KKSV20].…”

Section: Introductionmentioning

confidence: 99%

“…[Ham11]). Our setting is essentially a G-equivariant version of the case of discrete quantum groups studied recently in [KKSV20].…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles

Habbestad¹,

Hataishi²,

Neshveyev³

2021

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We clarify the relation between noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of quantum groups. Specifically, given a compact quantum group G, we show that in many cases where the Poisson boundary of the dual discrete quantum group Ĝ has been computed, the underlying topological boundary either coincides with the Furstenberg-Hamana boundary of the Drinfeld double D(G) of G or is a quotient of it. This includes the q-deformations of compact Lie groups, free orthogonal and free unitary quantum groups, quantum automorphism groups of finite dimensional C * -algebras. In particular, the boundary of D(Gq) for the q-deformation of a compact connected semisimple Lie group G is Gq/T (for q = 1), in agreement with the classical results of Furstenberg and Moore on the Furstenberg boundary of G C .We show also that the construction of the Furstenberg-Hamana boundary of D(G) respects monoidal equivalence and, in fact, can be carried out entirely at the level of the representation category of G. Along the way we categorify equivariant completely positive and completely bounded maps, thereby generalizing the notions of invariant means and multipliers for rigid C * -tensor categories.

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Noncommutative Furstenberg boundary

Cited by 5 publications

References 22 publications

Topological boundaries of connected graphs and Coxeter groups

Topological boundaries of connected graphs and Coxeter groups

On Ideals of $L^1$-algebras of Compact Quantum Groups

Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles

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