We conjecture that a countable group G admits a nonsingular Bernoulli action of type III 1 if and only if the first L 2 -cohomology of G is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III 1 Bernoulli actions of the groups Z and the free groups F n , with different degrees of ergodicity.

By a construction of Vaughan Jones, the bipartite graph Γ(A) associated with the natural inclusion of C inside a finite-dimensional C * -algebra A gives rise to a planar algebra P Γ(A) . We prove that every subfactor planar subalgebra of P Γ(A) is the fixed point planar algebra of a uniquely determined action of a compact quantum group G on A. We use this result to introduce a conceptual framework for the free wreath product operation on compact quantum groups in the language of planar algebras/standard invariants of subfactors. Our approach will unify both previous definitions of the free wreath product due to Bichon and Fima-Pittau and extend them to a considerably larger class of compact quantum groups. In addition, we observe that the central Haagerup property for discrete quantum groups is stable under the free wreath product operation (on their duals).Recall that π 0 (resp. π 1 ) is the pair partition of 2k with blocks {(2i, 2i + 1)} (resp. {2i + 1, 2i + 2)}). Definition 6.7. A free pair (T, T ) of planar tangles is called reduced if T = T π and T = T kr (π) for some non-crossing partition π such that π ≥ π 0 .An example of reduced free pair is given in Figure 17 with π = {{1, 6}, {2, 3, 4,

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica–Speicher [5] and Weber [42]. For the infinite‐dimensional Temperley–Lieb algebra (corresponding to the quantum group ON+$O^+_N$) and the Motzkin algebra (BN+$B^+_N$), the classification of extremal traces implies a classification result for well‐known types of central random lattice paths. For the 2‐Fuss–Catalan algebra (HN+$H_N^+$), we solve the classification problem by computing the minimal or exit boundary (also known as the absolute) for central random walks on the Fibonacci tree, thereby solving a probabilistic problem of independent interest, and to our knowledge the first such result for a nonhomogeneous tree. In the course of this article, we also discuss the branching graphs for all seven examples of free partition quantum groups, compute those that were not already known, and provide new formulae for the dimensions of their irreducible representations.

In this paper, we study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products G ≀ * S + N , where G is a compact matrix quantum group. Based on recent result on their corepresentation theory by Lemeux and Tarrago in [LemTa], we prove that G ≀ * S + N is of Kac type whenever G is, and that the reduced version of G ≀ * S + N is simple with unique trace state whenever N ≥ 8. Moreover, we prove that the reduced von Neumann algebra of G ≀ * S + N does not have property Γ.

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