Jean Renault
After a review of some of the main results about hyperfinite equivalence relations and their cocycles in the measured setting, we give a definition of a topological AF-equivalence relation. We show that every cocycle is cohomologous to a quasi-product cocycle. We then study the problem of determining the quasi-invariant probability measures admitting a given cocycle as their Radon-Nikodym derivative.
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In a recent article of Kenny De Commer, was investigated a Morita equivalence between locally compact quantum groups, in which a measured quantum groupoid, of basis $\mathbb{C}^2$, was constructed as a linking object. Here, we generalize all these constructions and concepts to the level of measured quantum groupoids. As for locally compact quantum groups, we apply this construction to the deformation of a measured quantum groupoid by a 2-cocycle.
We prove cocycle and orbit equivalence superrigidity for lattices in SL(n,R) acting linearly on R^n, as well as acting projectively on certain flag manifolds, including the real projective space. The proof combines operator algebraic techniques with the property (T) in the sense of Zimmer for the action of SL(n,Z) on R^n, n \geq 4. We also show that the restriction of the associated orbit equivalence relation to a subset of finite Lebesgue measure, provides a II_1 equivalence relation with property (T) and yet fundamental group equal to R_+.
We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.
Measured inverse semigroups and their actions on von Neumann algebras and equivalence relations
本文通过逆半群与 groupoids 的对应关系，统一证明了 von Neumann 代数包含和等价关系包含的陪集作用结果，填补了文献中的空白。
We prove that if a countable discrete group $Γ$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $Γ= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $Γ= H \times H'$ with $H$ an infinite Kazhdan group and $H'$ arbitrary), and $\Cal V$ is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. $\Cal V$ countable discrete, or separable compact), then any $\Cal V$-valued measurable cocycle for a measure preserving action $Γ\curvearrowright X$ of $Γ$ on a probability space $(X,μ)$ which is weak mixing on $H$ and {\it s-malleable} …
Proper cocycles, measure equivalence and $L_p$-Fourier multipliers
本文提出一种基于proper cocycles的传递方法，推广了$L_p$-Fourier乘子的结果，简化了非交换傅里叶级数收敛证明，并建立了Jodeit定理的非交换类比，构造了$SL_2(\mathbb{R})$上的希尔伯特变换。
Let R a be countable ergodic equivalence relation of type II_1 on a standard probability space (X,m). The group Out(R) of outer automorphisms of R consists of all invertible Borel measure preserving maps of the space which map R-classes to R-classes modulo those which preserve almost every R-class. We analyze the group Out(R) for relations R generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of Out(R) and explicitly computing this group for the standard actions. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's …
We introduce a generalization of the notion of approximately proper equivalence relations studied by Renault and with it we build an étale groupoid. Choosing a suitable set of continuous functions to play the role of a potential, we construct a cocycle in that groupoid and discuss the corresponding Radon-Nikodym problem.
We construct a group measure space II$_1$ factor that has two non-conjugate Cartan subalgebras. We show that the fundamental group of the II$_1$ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is non-trivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.
Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form $L(\mathcal{R})$ for countable probability measure preserving equivalence relations $\mathcal{R}$. We show that $L(\mathcal{R})$ is prime whenever $\mathcal{R}$ is nonamenable, ergodic, and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. This is accomplished by constructing the $\textit{Gaussian extension }\tilde{\mathcal{R}}$ of $\mathcal{R}$ and subsequently an s-malleable deformation of the inclusion $L(\mathc…
The group of automorphisms of the Cuntz algebra $\mathcal{O}_{2}$ is a Polish group with respect to the topology of pointwise convergence in norm. Our main result is that the relations of conjugacy and cocycle conjugacy of automorphisms of $\mathcal{O}_{2}$ are complete analytic sets and, in particular, not Borel. Moreover, we show that from the point of view of Borel complexity theory, classifying automorphisms of $\mathcal{O}_{2}$ up to conjugacy or cocycle conjugacy is strictly more difficult than classifying up to isomorphism any class of countable structures with Borel isomorphism relatio…
Strongly ergodic equivalence relations: spectral gap and type III invariants
本文利用谱隙判据刻画了标准测度空间上的强遍历等价关系，并证明了局部紧阿贝尔群可测1-上闭链导出的斜积等价关系的强遍历性，同时引入并研究了III型强遍历等价关系的Sd和τ不变量及其结构定理。
The residue cocycle associated to a suitable spectral triple is the key component of the Connes-Moscovici local index theorem in noncommutative geometry. We review the relationship between the residue cocycle and heat kernel asymptotics. We use a modified version of the Getzler calculus to compute the cocycle for a class of Dirac-type operators introduced by Bismut, obtained by deforming a Dirac operator by a closed 3-form B. We also compute the cocycle in low-dimensions when the 3-form B is not closed.
In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.
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