Stefaan Vaes, Nikolas Vander Vennet
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.
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Given a free unitary quantum group $G=A_u(F)$, with $F$ not a unitary $2$-by-$2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-$\hat G$-invariant, irreducible, finite range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.
Variations on quantum de Finetti theorems and operator valued Martin boundaries: a Choquet-Deny approach
本文通过引入量子群上的算子值 Martin 边界理论，以 Choquet-Deny 方法重新证明量子 de Finetti 定理及其变体，并推广了 Biane 定理。
In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of non-commutative operators; non-commutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these non-commutative algebras are the Cuntz algebra…
Space-time boundaries for random walks and their application to operator algebras
本文研究了随机游走的空间-时间马丁边界，并将其与算子代数中的紧致化相关联，证明了0-马丁边界与λ-马丁核的联系，揭示了非交换谢洛夫边界与Toeplitz C*代数的一致性。
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 Poiss…
A characterization of $C^*$-simplicity of countable groups via Poisson boundaries
该研究通过二分法刻画可数群的$C^*$-单性：$C^*$-单群对一般测度的泊松边界作用本质自由，非$C^*$-单群则相反，建立了$C^*$-单性与泊松边界作用的本质自由性的关联。
Noncommutative Poisson boundaries, ultraproducts and entropy
本文利用 von Neumann 代数超积构造带迹情形的非交换 Poisson 边界，并据此完善 Kaimanovich-Vershik 非交换熵基本定理，证明两类边界等价关系。
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…
Non-C*-simple groups admit non-free actions on their Poisson boundaries
本文证明每个可数非C*-单群都存在一个全支撑对称测度，使其Poisson边界上的作用具有非平凡稳定化子，从而为具有平凡顺从根的非C*-单群构造了大量非正规稳定化子的非自由作用例子。
Peripheral Poisson boundaries and jointly bi-harmonic functions
本研究解答了Kaimanovich关于可数离散群上联合双调和函数的问题，解决了Bhat等人关于对称生成测度诱导Markov算子的外围特征值猜想，完整刻画了对应的外围Poisson边界。
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.
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.
Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles
该研究阐明量子群的非交换泊松边界与Drinfeld double的Furstenberg-Hamana边界的关系，证明其在多种量子群情形下的重合或商集关系，还将该边界构造推广至刚性C$^*$-张量范畴。
On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices
本文证明，对于实连通半单李群中的不可约格，其最小抛物子群交叉积中的不变冯·诺依曼子代数必为特定形式，证实了Amrutam-Hartman猜想。
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…
Entropies and Poisson boundaries of random walks on groups with rapid decay
本文研究随机游走群的熵与泊松边界，通过调和分析和Banach空间理论方法，证明了Lyapunov指数与谱半径渐进行为的关系，并展示了Avez熵与Furstenberg熵的等价性，以及快速衰减群的卷积熵与Avez熵的一致性。
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.
A classical construction associates to a transient random walk on a discrete group $Γ$ a compact $Γ$-space $\partial_M Γ$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M Γ) \rtimes_r Γ$ comes equipped with a one-parameter group of automorphisms given by the Martin kernels that define the Martin boundary. In this paper we study the KMS states for this flow and obtain a complete description when the Poisson boundary of the random walk is trivial and when $Γ$ is a torsion free non-elementary hyperbolic group. We also construct examples to show that the stru…
The Martin boundary of a discrete quantum group
本文研究离散量子群上由q-迹态卷积定义的马尔可夫算子的Martin边界，建立正调和元的表示定理，构造类似泊松积分的u.c.p.映射，并将SU_q(2)对偶的Martin边界等同于Podleś量子齐性球面，推广统一了前人结果。
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