Standard Measure Space Research Articles

Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $\sigma $-finite standard measure space $

Strongly ergodic equivalence relations: spectral gap and type III invariants

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

A remark on fullness of some group measure space von Neumann algebras

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

Ergodic properties of skew products in infinite measure

Let (\Omega,\mu) be a shift of finite type with a Markov probability, and (Y,\nu) a non-atomic standard measure space. For each symbol i of the symbolic space, let \Phi_i be a measure-preserving automorphism of (Y,\nu). We study skew products of the form (\omega,y) --> (\sigma\omega,\Phi_{\omega_0}(y)), where \sigma =shift map on (\Omega,\mu). We prove that, when the skew product is conservative, it is ergodic if and only if the \Phi_i's have no common non-trivial invariant set. In the second part we study the skew product when \Omega={0,1}^Z, \mu =Bernoulli measure, and \Phi_0,\Phi_1 are R-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to \sqrt{n}, and its trajectories satisfy the central, functional central and local limit theorem.

Amalgamated free product type III factors with at most one Cartan subalgebra

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

Skorohod's Representation Theorem for Sets of Probabilities

From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zero-one if there exists an E in F, a measurable, onto phi:Omega -> Pi such that for all p in Pi, p(phi^{-1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space,
Delta_M is the set of Borel probabilities on M and Comp(
Delta_M) is the class of non-empty, compact subsets of Delta_
M with the Hausdor ff metric. There exists a jointly measurable H: Comp(
Delta_M) x Omega ->M such that for all K in Comp(Delta_
M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) -->0, then for all p in Pi, p({omega: H(K_n,omega) -->H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's representation theorem.

Measurable chromatic and independence numbers for ergodic graphs and group actions

We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan’s property (T), and freeness. We also prove a Borel analog of the classical Brooks’ Theorem in finite combinatorics for actions of groups with finitely many ends.

A survey on spectral multiplicities of ergodic actions

AbstractGiven a transformationTof a standard measure space (X,μ), let ℳ(T) denote the set of spectral multiplicities of the Koopman operatorUTdefined in$L^2(X,\mu )\ominus \Bbb C$byUTf:=f∘T. In this survey paper we discuss which subsets of$\Bbb N\cup \{\infty \}$are realizable as ℳ(T) for variousT: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure-preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.

Closable Multipliers

Let (X, μ) and (Y, ν) be standard measure spaces. A function $${\varphi\in L^\infty(X\times Y,\mu\times\nu)}$$ is called a (measurable) Schur multiplier if the map S φ , defined on the space of Hilbert-Schmidt operators from L 2(X, μ) to L 2(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S φ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if φ is of Toeplitz type, that is, if φ(x, y) = f(x − y), $${x,y\in G}$$ , where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) − f(y))/(x − y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.

One-cocycles on smooth flows of weights and extended modular coactions

This paper is concerned with Borel 1-cocycles on ergodic flows on standard Borel measure spaces and a certain type of group coactions on (separable) factors.

Operators Represented by Conditional Expectations and Random Measures

On standard measure spaces every order continuous linear map between two ideals of almost everywhere finite measurable functions can be represented by a random measure. An analogue of this theorem is proved for the case of arbitrary σ-finite measure spaces. This fact leads to a proof that every order continuous linear map between ideals of almost everywhere finite measurable functions on σ-finite measure spaces is multiplication conditional expectation representable. This sheds further light on the structure of order continuous operators.

On subrelations of ergodic measured type III equivalence relations

We discuss the classification up to orbit equivalence of inclusions S ⊂ R of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of S ⊂ R), an ergodic nonsingular R-flow V and a homomorphism of G to the centralizer of V . 0. Introduction. We consider nonsingular discrete ergodic hyperfinite equivalence relations on a standard measure space. Our concern is to classify pairs (R,S) where R is an ergodic equivalence relation and S ⊂ R is a subrelation of finite index (which means that the R-equivalence class of a.e. point consists of finitely many S-classes), up to orbit equivalence. This problem is closely related to the classification of subfactors in von Neumann algebras theory. For a single equivalence relation R the problem was solved by H. Dye [Dy] and W. Krieger [Kr] in terms of the associated flows. Then, in the case where R is of type II1, J. Feldman, C. Sutherland, and R. Zimmer [FSZ] provided a simple classification of ergodic R-subrelations of finite index and normal R-subrelations of arbitrary index. (We remark that in an earlier paper [Ge] M. Gerber classified R-subrelations of finite index in a different—but equivalent—context of finite extensions of ergodic probability preserving transformations.) These results were further extended in [Da1, §4] and [Da2], where quasinormal subrelations of type II1 were introduced and studied. Recently, T. Hamachi considered finite index subrelations of a type III0 equivalence relation R, introduced a system of invariants for orbit equiva2000 Mathematics Subject Classification: 28D99, 46L55. The research was supported in part by INTAS 97-1843.

The Inverse Stable Range Functor

We give an inverse construction of the stable range for general flows which may or may not admit an invariant measure. The inverse map is then shown to be a right inverse functor of the stable range functor. The Krieger theorem asserts that in the type IIIo case, there is a one-toone correspondence between orbit equivalent classes of ergodic transformations, algebraic isomorphism classes of Krieger factors, and conjugate isomorphism classes of nontransitive ergodic flows. This correspondence is established via explicit maps between the objects, namely the stable range map, the crossed product, and the flow of weights. In this paper we are concerned with inverting the stable range map, i.e., given a flow on a standard measure space, we will explicitly construct a transformation with the flow as its stable range. Since orbit equivalent transformations give rise to conjugate flows under the stable range map, the choice of the transformation is not unique. However, by using some kind of skew product with the odometer, it is possible to choose a natural one which satisfies the requirement. This construction provides the vital missing arrow which points in the opposite direction of the other maps. For the case where the flow admits an invariant measure, such a transformation can be found in [4] or [5], but for a general flow the problem is more difficult. In [3] Hamachi showed that for a given ergodic flow, one can construct an ergodic action of Z x (Z + aZ) on a certain standard measure space whose stable range is isomorphic to the flow. The result of Connes, Feldman, and Weiss guarantees that the equivalence relation generated by this action is generated by a single transformation T, but it is far from clear how to construct such a transformation explicitly. Noting that the stable range map is in fact a functor from the category of transformations with orbit transporting isomorphisms to the category of flows with conjugations, we see that the construction below also extends to the construction of a right inverse functor to the stable range functor. This fact establishes that the mod homomorphism from the automorphism group of a measured equivalence relation to the automorphism group of its associated flow is split (this also follows from Hamachi's construction [3, p. 399]). Our method also yields a transparent proof of surjectivity of mod, a fact noted by both Hamachi and Golodets [1]. Received by the editors June 8, 1990 and, in revised form, November 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D99. ? 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

A nonstandard solution of the korteweg-de vries equation

A few years ago, L. Arkeryd solved the problem of the existence of a solution global in time of the nonlinear Boltzmann equation far from equilibrium by the use of nonstandard analysis (see [2,3], [5]). The purpose of this work — which is an abridged version of the thesis [9] — is to adapt Arkeryd's idea to the Korteweg- de Vries equation. In short, the program consists of three parts: • Proof of the existence of a solution for a truncated KdV equation; • Shifting away the truncation to infinity by use of the transfer principle; • Pulling back the solution to the standard world by an application of P.A. • Loeb's measure-theoretic construction of a standard measure space from an internal one. Up to now, only two ways of solving the KdV have been known: the explicit construction of solitons by using the special wave propagation ansatz Ф ≡ Ф(x−vt) and the inverse scattering method. The nonstandard solution has the advantage that no special ansatz is needed. Moreover, it is not a complicated implicit construction but a straightforward existence proof. The article involves some basic concepts of nonstandard analysis which can be found in [1], [6] or [7]; except for that it is essentially self-contained.

Structure of mixing and category of complete mixing for stochastic operators

Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measu

A Borel Parametrlzatlon of Polish Groups

This paper constructs a standard space, PG, and a map p^PG—>G(p} such that each G(p) is a Polish group, and such that every Polish group is isomorphic to at least one of the groups G(p) ; PG thus serves as a parameter space for all Polish groups. We formulate the notion of a map from a standard B Space to Polish groups, and that of a from a standard groupoid to Polish groups; both are defined in terms of the existence of factorizations through PG. We apply these ideas to establish a general Cohomology Lemma, asserting that cocycles, with values in family of Polish groups, may be cobounded into a given family of dense, normal, subgroups, whenever the underlying groupoid is a hyperfinite equivalence relation. The purpose of this paper is to provide a parametrizatio n, by a standard space PG, for the space of Polish topological groups, i. e. those second countable topological groups whose underlying topology may be defined by a complete metric, and to present applications of this to the notion of Borel functor from a standard groupoid to Polish groups. The need for such concepts became apparent during the course of joint work with M. Takesaki on the classification of the possible actions (up to cocycle conjugacy) of a discrete amenable group on a hyperfinite, semifinite injective von Neumann algebra [12], and the paper can be viewed as preparatory to this work. However, the point of view adopted also reveals a definition of A. Connes, [4], of the notion of from a standard groupoid to standard measure spaces, as being very natural. A common situation in which the problems considered here arise is the following: if G is a Polish group and X a Polish G-space under

Peter A. Loeb. Conversion from nonstandard to standard measure spaces and applications in probability theory. Transactions of the American Mathematical Society, vol. 211 (1975), pp. 113–122. - Robert M. Anderson. A non-standard representation for Brownian motion and ltô integration. Israel journal of mathematics, vol. 25 (1976), pp. 15–46.

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Coboundaries and Homomorphisms for Non-Singular Actions and a Problem of H. Helson

Let f be a one-cocycle for a non-singular action of a locally compact group G on a standard measure space (Y, μ) with values in a locally compact abelian group A. If χ ɛ Â, χ(f) is a one-cocycle with values in the circle group T. We investigate the question of when one can conclude that f is a coboundary, given that χ(f) is a coboundary for some ‘sufficiently large’ set of χ. We obtain very precise results, extending previous work of Hamachi, Oka, and Osikawa. We also show that if G is measure-preserving, and  is connected, f is a coboundary if and only if it is ‘bounded’ in a certain natural sense. We then investigate a companion problem posed by H. Helson to determine whether one can conclude that f is cohomologous to a homomorphism of G into A if the same is known to be true for all or for sufficiently many χ(f)'s. We show that the answer is affirmative in some cases of interest but is negative in general, and, in fact, compute the defect. The counter-examples produced are examples of non-singular integer actions which have certain prescribed uncountable groups of eigenfunctions; and hence may be of some independent interest.

Pseudo-Integral Operators

Let $(X, \mathcal {a}, m)$ be a standard finite measure space. A bounded operator T on ${L^2}(X)$ is called a pseudo-integral operator if $(Tf)(x) = \int {f(y) \mu (x, dy)}$, where, for every x, $\mu (x, \cdot )$ is a bounded Borel measure on X. Main results: 1. A bounded operator T on ${L^2}$ is a pseudo-integral operator with a positive kernel if and only if T maps positive functions to positive functions. 2. On nonatomic measure spaces every operator unitarily equivalent to T is a pseudo-integral operator if and only if T is the sum of a scalar and a Hilbert-Schmidt operator. 3. The class of pseudo-integral operators with absolutely bounded kernels form a selfadjoint (nonclosed) algebra, and the class of integral operators with absolutely bounded kernels is a two-sided ideal. 4. An operator T satisfies $(Tf)(x) = \int {f(y) \mu (x, dy)}$ for $f \in {L^\infty }$ if and only if there exists a positive measurable (almost-everywhere finite) function $\Omega$ such that $\left | {(Tf)(x)} \right | \leqslant {\left \| f \right \|_\infty }\Omega (x)$.

Pseudo-integral operators

Let ( X , a , m ) (X,\,\mathcal {a},\,m) be a standard finite measure space. A bounded operator T on L 2 ( X ) {L^2}(X) is called a pseudo-integral operator if ( T f ) ( x ) = ∫ f ( y ) μ ( x , d y ) (Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)} , where, for every x, μ ( x , ⋅ ) \mu (x,\, \cdot \,) is a bounded Borel measure on X. Main results: 1. A bounded operator T on L 2 {L^2} is a pseudo-integral operator with a positive kernel if and only if T maps positive functions to positive functions. 2. On nonatomic measure spaces every operator unitarily equivalent to T is a pseudo-integral operator if and only if T is the sum of a scalar and a Hilbert-Schmidt operator. 3. The class of pseudo-integral operators with absolutely bounded kernels form a selfadjoint (nonclosed) algebra, and the class of integral operators with absolutely bounded kernels is a two-sided ideal. 4. An operator T satisfies ( T f ) ( x ) = ∫ f ( y ) μ ( x , d y ) (Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)} for f ∈ L ∞ f\, \in \,{L^\infty } if and only if there exists a positive measurable (almost-everywhere finite) function Ω \Omega such that | ( T f ) ( x ) | ⩽ ‖ f ‖ ∞ Ω ( x ) \left | {(Tf)(x)} \right |\, \leqslant \,{\left \| f \right \|_\infty }\Omega (x) .