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Since the early days of Tomita–Takesaki theory, it is known that a von Neumann algebra 푀 that admits a state 휑 with trivial centralizer M φ must be a type III1 factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense G δ set among all faithful normal states on any III1 factor with separable predual. Through Connes' Radon–Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper. [ABSTRACT FROM AUTHOR]

We construct for every connected locally finite graph |$\Pi $| the quantum automorphism group |$\operatorname{QAut} \Pi $| as a locally compact quantum group. When |$\Pi $| is vertex transitive, we associate to |$\Pi $| a new unitary tensor category |${\mathcal{C}}(\Pi)$| and this is our main tool to construct the Haar functionals on |$\operatorname{QAut} \Pi $|⁠. When |$\Pi $| is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs |$\Pi $|⁠ , |$\Pi ^{\prime}$| and prove that this implies monoidal equivalence of |$\operatorname{QAut} \Pi $| and |$\operatorname{QAut} \Pi ^{\prime}$|⁠. [ABSTRACT FROM AUTHOR]

We undertake a systematic study of W ∗ -rigidity paradigms for the embeddability relation ↪ between separable II 1 factors and its stable version ↪ s , obtaining large families of non stably isomorphic II 1 factors that are mutually embeddable and families of II 1 factors that are mutually non stably embeddable. We provide an augmentation functor G ↦ H G from the category of groups into icc groups, so that L (H G 1) ↪ s L (H G 2) iff G 1 ↪ G 2 . We construct complete intervals of II 1 factors, including a strict chain of II 1 factors (M k) k ∈ Z with the property that if N is any II 1 factor with M i ↪ s N and N ↪ s M j , then N ≅ M k t for some i ≤ k ≤ j and t > 0 . [ABSTRACT FROM AUTHOR]

Given a topologically free action of a countable group G on a compact metric space X, there is a canonical correspondence between continuous 1-cocycles for this group action and diagonal 1-parameter groups of automorphisms of the reduced crossed product C ∗ -algebra. The KMS spectrum is defined as the set of inverse temperatures for which there exists a KMS state. We prove that the possible KMS spectra depend heavily on the nature of the acting group G. For groups of subexponential growth, we prove that the only possible KMS spectra are { 0 } , [ 0 , + ∞) , (- ∞ , 0 ] and R . For certain wreath product groups, which are amenable and of exponential growth, we prove that any closed subset of R containing zero arises as KMS spectrum. Finally, for certain nonamenable groups including the free group with infinitely many generators, we prove that any closed subset may arise. Besides uncovering a surprising relation between geometric group theoretic properties and KMS spectra, our results provide two simple C ∗ -algebras with the following universality property: any closed subset (containing, resp. not containing zero) arises as the KMS spectrum of a 1-parameter group of automorphisms of this C ∗ -algebra. [ABSTRACT FROM AUTHOR]

Given a topologically free action of a countable group G on a compact metric space X, there is a canonical correspondence between continuous 1-cocycles for this group action and diagonal 1-parameter groups of automorphisms of the reduced crossed product C ∗ -algebra. The KMS spectrum is defined as the set of inverse temperatures for which there exists a KMS state. We prove that the possible KMS spectra depend heavily on the nature of the acting group G. For groups of subexponential growth, we prove that the only possible KMS spectra are { 0 } , [ 0 , + ∞) , (- ∞ , 0 ] and R . For certain wreath product groups, which are amenable and of exponential growth, we prove that any closed subset of R containing zero arises as KMS spectrum. Finally, for certain nonamenable groups including the free group with infinitely many generators, we prove that any closed subset may arise. Besides uncovering a surprising relation between geometric group theoretic properties and KMS spectra, our results provide two simple C ∗ -algebras with the following universality property: any closed subset (containing, resp. not containing zero) arises as the KMS spectrum of a 1-parameter group of automorphisms of this C ∗ -algebra. [ABSTRACT FROM AUTHOR]

It is a wide open problem to give an intrinsic criterion for a II1 factor M to admit a Cartan subalgebra A. When A ⊂ M is a Cartan subalgebra, the A-bimodule L²(M) is simple in the sense that the left and right actions of A generate a maximal abelian subalgebra of B(L²(M)). A II1 factor M that admits such a subalgebra A is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II1 factorM to be s-thin and left open the question whether all s-thin II1 factors admit a Cartan subalgebra. We answer this question negatively by constructing s-thin II1 factors without Cartan subalgebras. [ABSTRACT FROM AUTHOR]

Abstract Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Żuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T). [ABSTRACT FROM AUTHOR]

We conjecture that a countable group G admits a nonsingular Bernoulli action of type III1 if and only if the first L2-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 III1 Bernoulli actions of the groups Z and the free groups Fn, with different degrees of ergodicity. [ABSTRACT FROM AUTHOR]

We construct the first II 1 factors having exactly two group measure space decompositions up to unitary conjugacy. Also, for every positive integer n , we construct a II 1 factor M that has exactly n group measure space decompositions up to conjugacy by an automorphism. [ABSTRACT FROM AUTHOR]

We develop a representation theory for $${\lambda}$$ -lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $${\lambda}$$ -lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with property (T) standard invariant and that are not constructed from property (T) groups. [ABSTRACT FROM AUTHOR]

Crossed products with noncommutative Bernoulli actions were introduced by Connes as the first examples of full factors of type I I I . This article provides a complete classification of the factors ( P , ϕ ) F n ⋊ F n , where F n is the free group and P is an amenable factor with an almost periodic state ϕ . We show that these factors are completely classified by the rank n of the free group F n and Connes's Sd-invariant. We prove similar results for free product groups, as well as for classes of generalized Bernoulli actions. [ABSTRACT FROM AUTHOR]

We prove that the continuous version of the Connes-Shlyakhtenko first L2-cohomology for II1 factors, as proposed by Thom, always vanishes. [ABSTRACT FROM AUTHOR]

We prove that for any free ergodic probability measure preserving action of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II 1 factor has as its unique Cartan subalgebra, up to unitary conjugacy. [ABSTRACT FROM AUTHOR]

Every affine isometric action α of a group G on a real Hilbert space gives rise to a nonsingular action α ˆ of G on the associated Gaussian probability space. In the recent paper [2] , several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation π of G is mixing. We develop new methods to prove ergodicity when π is only weakly mixing. We determine the type of α ˆ in full generality. Using Cantor measures, we give examples of type III 1 ergodic Gaussian actions of Z whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions. [ABSTRACT FROM AUTHOR]

We prove that for many nonamenable groups Γ, including all hyperbolic groups and all nontrivial free products, the left–right wreath product group ?? ≔ (ℤ/2ℤ)(Γ) ⋊ (Γ×Γ) is W*-superrigid. This means that the group von Neumann algebra L ?? entirely remembers ??. More precisely, if L ?? is isomorphic with L Λ for an arbitrary countable group Λ, then Λ must be isomorphic with ??. [ABSTRACT FROM PUBLISHER]

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$ of a free group on n generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$ , the associated group measure space II factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$ has L( X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products. [ABSTRACT FROM AUTHOR]

Abstract: We prove that for any free ergodic nonsingular nonamenable action of all Γ in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra has as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in Popa and Vaes (in press) [38]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products over a subalgebra B of type I. [Copyright &y& Elsevier]

Abstract: We construct a class of II1 factors that admit unclassifiably many Cartan subalgebras in the sense that the equivalence relation of being conjugate by an automorphism of is complete analytic, in particular non Borel. We also construct a II1 factor that admits uncountably many non isomorphic group measure space decompositions, all involving the same group . So is a group that admits uncountably many non stably orbit equivalent actions whose crossed product II1 factors are all isomorphic. [Copyright &y& Elsevier]

A probability measure preserving action Γ … (X, μ) is called rigid if the inclusion of L∞(X) into the crossed product L∞ (X)… Γ has the relative property (T) in the sense of Popa. We give examples of rigid, free, probability measure preserving actions that are ergodic but not strongly ergodic. The same examples show that rigid actions may admit non-rigid quotients. [ABSTRACT FROM AUTHOR]

We construct inner amenable groups G with infinite conjugacy classes and such that the associated II factor has no non-trivial asymptotically central elements, i.e. does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975. [ABSTRACT FROM AUTHOR]

We prove that for a fairly large family of HNN extensions Τ, the group measure space II1 factor L∞(X) x Τ given by an arbitrary free ergodic probability measure preserving action of Τ has a unique group measure space Cartan subalgebra up to unitary conjugacy. From this we deduce new examples of W*-superrigid group actions, i.e. where the II1 factor L∞(X) x Τ entirely remembers the group action from which it was constructed. [ABSTRACT FROM AUTHOR]

We prove a 'unique crossed product decomposition' result for group measure space II factors L ( X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family $\mathcal{G}$, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T denotes the group of upper triangular matrices in PSL ( n,ℤ), then any free, mixing p.m.p. action of $\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z})$ is W-superrigid, i.e. any isomorphism between L ( X)⋊Γ and an arbitrary group measure space factor L ( Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family $\mathcal{G}$, the Bernoulli actions of Γ are W-superrigid. [ABSTRACT FROM AUTHOR]

We identify the Poisson boundary of the dual of the universal compact quantum group Au(F) with a measurable field of ITPFI (infinite tensor product of finite type I) factors. [ABSTRACT FROM AUTHOR]

We prove that given any compact group G, there exists a minimal action of G on a II1 factor M such that the bimodule category of the fixed-point II1 factor MG is naturally equivalent with the representation category of G. In particular, all subfactors of MG with finite Jones index can be described explicitly. [ABSTRACT FROM AUTHOR]

The article examines the actions of F∞ whose II1 factors and orbit equivalence relations have prescribed fundamental group. A large family of S of subgroups of R+, containing R+ itself are exhibited. The first examples of II1 factors with separable predual having uncountable fundamental group different from R+. The ergodic measures and Hausdorff dimension are discussed.

Abstract: We show that any compact group can be realized as the outer automorphism group of a factor of type II1. This has been proved in the abelian case by Ioana, Peterson and Popa [A. Ioana, J. Peterson, S. Popa, Amalgamated free products of w-rigid factors and calculation of their symmetry group, math.OA/0505589, Acta Math., in press] applying Popa''s deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs. [Copyright &y& Elsevier]

Abstract: We study equivalence relations and factors associated with (quotients of) generalized Bernoulli actions of Kazhdan groups. Specific families of these actions are entirely classified up to isomorphism of factors. This yields explicit computations of outer automorphism and fundamental groups. In particular, every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic factors. [Copyright &y& Elsevier]

Abstract: Given a closed quantum subgroup of a locally compact quantum group, we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group. More generally, we study induction given a coaction of the quantum subgroup on a -algebra. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. This means that we define quantum homogeneous spaces as -algebras and that we prove Morita equivalence of crossed products and homogeneous spaces. We essentially use von Neumann algebraic techniques to prove these Morita equivalences between -algebras. [Copyright &y& Elsevier]

In the framework of locally compact quantum groups, we study cocycle actions. We develop the cocycle bicrossed product construction, starting from a matched pair of locally compact quantum groups. We define exact sequences and establish a one-to-one correspondence between cocycle bicrossed products and cleft extensions. In this way, we obtain new examples of locally compact quantum groups. [Copyright &y& Elsevier]

In this paper we define and study Hopf C$^*$-algebras. Roughly speaking, a Hopf C$^*$-algebra is a C$^*$-algebra $A$ with a comultiplication $\phi : A \rightarrow M(A \otimes A)$ such that the maps $a \otimes b \mapsto \phi(a)(1 \otimes b)$ and $a \otimes b \mapsto (a \otimes 1)\phi(b)$ have their range in $A \otimes A$ and are injective after being extended to a larger natural domain, the Haagerup tensor product $A \otimes_h A$. In a purely algebraic setting, these conditions on $\phi$ are closely related to the existence of a counit and antipode. In this topological context, things turn out to be much more subtle, but nevertheless one can show the existence of a suitable counit and antipode under these conditions.The basic example is the C$^*$-algebra $C_0(G)$ of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication. But also the reduced group C$^*$-algebra $C^*_r(G)$ of a locally compact group with the well-known comultiplication falls in this category. In fact all locally compact quantum groups in the sense of Kustermans and the first author (such as the compact and discrete ones) as well as most of the known examples are included.This theory differs from other similar approaches in that there is no Haar measure assumed.2000 Mathematics Subject Classification: 46L65, 46L07, 46L89. [ABSTRACT FROM AUTHOR]

We determine the Krieger type of nonsingular Bernoulli actions G ↷ ∏ g ∈ G ({ 0 , 1 } , μ g) . When G is abelian, we do this for arbitrary marginal measures μ g . We prove in particular that the action is never of type II ∞ if G is abelian and not locally finite, answering Krengel's question for G = Z . When G is locally finite, we prove that type II ∞ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II 1 or III 1 . When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of Vaes and Wahl (Geom Funct Anal 28:518–562, 2018) by proving that a group G admits a Bernoulli action of type III 1 if and only if G has nontrivial first L 2 -cohomology. [ABSTRACT FROM AUTHOR]

We prove that the regular von Neumann subalgebras B of the hyperfinite II 1 factor R satisfying the condition B ′ ∩ R = Z (B) are completely classified (up to conjugacy by an automorphism of R) by the associated discrete measured groupoid G = G B ⊂ R. We obtain a similar classification result for triple inclusions A ⊂ B ⊂ R , where A is a Cartan subalgebra in R and the intermediate von Neumann algebra B is regular in R. A key step in proving these results is to show the vanishing cohomology for the associated cocycle actions (α B ⊂ R , u B ⊂ R) of G on B. We in fact prove two very general vanishing cohomology results for free cocycle actions (α , u) of amenable discrete measured groupoids G on arbitrary tracial von Neumann algebras B , resp. Cartan inclusions A ⊂ B. Our work provides a unified approach and generalizations to many known vanishing cohomology and classification results [6] , [25] , [35] , [3] , [10] , [29] , etc. [ABSTRACT FROM AUTHOR]

We obtain a complete classification of a large class of non-almost-periodic free Araki-Woods factors Γ(μ, m)" up to isomorphism. We do this by showing that free Araki-Woods factors Γ(μ, m)" arising from finite symmetric Borel measures μ on ℝ whose atomic part μa is nonzero and not concentrated on {0} have the joint measure class C(Vk≥1 μ*k) as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras. [ABSTRACT FROM AUTHOR]

We show that Shlyakhtenko's free Araki-Woods factors are strongly solid, meaning that for any diffuse amenable von Neumann subalgebra that is the range of a normal conditional expectation, the normalizer remains amenable. This provides the first class of nonamenable strongly solid type III factors. [ABSTRACT FROM AUTHOR]

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa’s class S. [ABSTRACT FROM AUTHOR]

We introduce L²-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups #915;, we recover the ordinary (co)homology of #915;. For Cartan subalgebras, we recover Gaboriau's L²-Betti numbers for the associated equivalence relation. In this common framework, we prove that the L²-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property, and amenability. We compute the L²-Betti numbers for the standard invariants of the Temperley--Lieb--Jones subfactors and of the Fuss--Catalan subfactors, as well as for free products and tensor products. [ABSTRACT FROM AUTHOR]

We prove that the L²-Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L²-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and unreduced L²-Betti numbers of G agree and that the L²-Betti numbers of a lattice Γ in G equal those of G up to scaling by the covolume of Γ in G. We also deduce several vanishing results, including the vanishing of the reduced L²-cohomology for amenable locally compact groups. [ABSTRACT FROM AUTHOR]

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood. [ABSTRACT FROM AUTHOR]

: In this paper, we study C*-algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi-regular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C*-algebras. On the way, we provide several remarks on C*-algebraic properties of quantum groups and their actions. [ABSTRACT FROM AUTHOR]