We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet--Deny theorem holds for compact quantum groups; also, the result of Kaimanovich--Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a non-commutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group $SU_{q}(2)$ arising from measures on its spectrum.
We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ factor into its boundary we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing…
Let $G$ be a co-amenable compact quantum group. We show that a right coideal of $G$ is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson bo…
For a locally compact quantum group $\mathbb{G}$, consider the convolution action of a quantum probability measure $μ$ on $L_\infty(\mathbb{G})$. As shown by Junge--Neufang--Ruan, this action has a natural extension to a Markov map on $\mathcal{B}(L_2(\mathbb{G}))$. We prove that the Poisson boundary of the latter can be realized concretely as the von Neumann crossed product of the Poisson boundary associated with $μ$ under the action of $\mathbb{G}$ induced by the coproduct. This yields an affirmative answer, for general locally compact quantum groups, to a problem raised by Izumi (2004) in t…
Actions of locally compact groups and quantum groups on W*-ternary rings of operators are discussed and related crossed products introduced. The results generalise those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed point spaces for convolution operators generated by contractive, non-necessarily positive measures, both in the classical and in the quantum context.
The Poisson and Martin boundaries for invariant random walks on the dual of the orthogonal quantum groups A_o(F), are identified with higher dimensional Podles spheres that we describe in terms of generators and relations. This provides the first such identification for random walks on non-amenable discrete quantum groups.
Bichon, De Rijdt and Vaes introduced the notion of monoidally equivalent compact quantum groups. In this paper we prove that there is a natural bijective correspondence between actions of monoidally equivalent quantum groups on unital C*-algebras or on von Neumann algebras. We apply this correspondence to study the behavior of Poisson and Martin boundaries under monoidal equivalence of quantum groups.
Given a rigid C*-tensor category C with simple unit and a probability measure $μ$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(C,μ)$. This is a new C*-tensor category P, generally with nonsimple unit, together with a unitary tensor functor $Π: C \to P$. Our main result is that if P has simple unit (which is a condition on some classical random walk), then $Π$ is a universal unitary tensor functor defining the amenable dimension function on C. Corollaries of this theorem unify various results in the literature on amenability of C*-tensor categorie…
By working with several specific Poisson-Lie groups arising from Heisenberg Lie bialgebras and by carrying out their quantizations, a case is made for a useful but simple method of constructing locally compact quantum groups. The strategy is to analyze and collect enough information from a Poisson-Lie group, and using it to carry out a ``cocycle bicrossed product construction''. Constructions are done using multiplicative unitary operators, obtaining C*-algebraic, locally compact quantum (semi-)groups.
We use compactifications of C*-algebras to introduce noncommutative coarse geometry. We transfer a noncommutative coarse structure on a C*-algebra with an action of a locally compact Abelian group by translations to Rieffel deformations and prove that the resulting noncommutative coarse spaces are coarsely equivalent. We construct a noncommutative coarse structure from a cocompact continuously square-integrable action of a group and show that this is coarsely equivalent to the standard coarse structure on the group in question. We define noncommutative coarse maps through certain completely po…
Beginning with a skew-symmetric matrix, we define a certain Poisson--Lie group. Its Poisson bracket can be viewed as a cocycle perturbation of the linear (or "Lie-Poisson") Poisson bracket. By analyzing this Poisson structure, we gather enough information to construct a C*-algebraic locally compact quantum group, via the "cocycle bicrossed product construction" method. The quantum group thus obtained is shown to be a deformation quantization of the Poisson-Lie group, in the sense of Rieffel.
A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary represen…
Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$, we show that any positive, possibly unbounded, harmonic function on $H$ with respect to an irreducible invariant random walk is $G$-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of $H$ coincide with those of $H/G$. A similar result is also proved in the setting of exact sequences of C$^*$-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.
