Perturbations of Subalgebras of Type ${\rm {II}}_1$ Factors
Sorin Popa, Allan Sinclair, Roger Smith
备注：
This submission corrects both an error in the statement of Lemma 2.1(ii) and an induced error in the proof of Theorem 5.2, the statement of which remains the same. These were discovered after publication, so a correction will appear in J. Funct. Anal
We consider two von Neumann subalgebras $\cl B_0$ and $\cl B$ of a type ${\rm{II}}_1$ factor $\cl N$. For a map $ϕ$ on $\cl N$, we define \[\|ϕ\|_{\infty,2}=\sup\{\|ϕ(x)\|_2\colon \|x\| \leq 1\},\] and we measure the distance between $\cl B_0$ and $\cl B$ by the quantity $\|{\bb E}_{\cl B_0}-{\bb E}_{\cl B}\|_{\infty,2}$. Under the hypothesis that the relative commutant in $\cl N$ of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the $\|\cdot \|_2$--norm. This hypothesis is satisfied, in particular, by masas and subfactors of trivial relative commutant. A general version with a slightly weaker conclusion is also proved. As a consequence, we show that if $\cl A$ is a masa and $u\in\cl N$ is a unitary such that $\cl A$ and $u\cl Au^*$ are close, then $u$ must be close to a unitary which normalizes $\cl A$. These qualitative statements are given quantitative formulations in the paper.
We study uniform perturbations of intermediate C*-subalgebras of inclusions of simple C*-algebras. If a unital simple C*-algebra has a simple C*-subalgebra of finite index, then sufficiently close simple intermediate C*-subalgebras are unitarily equivalent. These C*-subalgebras need not to be nuclear. The unitary can be chosen in the relative commutant algebra. An imediate corollary is the following: If the relative commutant is trivial, then the set of intermediate C*-subagebras is a finite set.
In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let N and M be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner-Popa. If N and M are sufficiently close, then N and M are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.
This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors.
It is a wide open problem to give an intrinsic criterion for a II_1 factor $M$ to admit a Cartan subalgebra $A$. When $A \subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is "simple" in the sense that the left and right action of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A II_1 factor $M$ that admits such a subalgebra $A$ is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II_1 factor $M$ to be s-thin and left open the question whether all s-thin II_1 factors admit a Cartan subalgebra. We answer this question negatively by construc…
We show that any amenable von Neumann subalgebra of any free Araki-Woods factor that is globally invariant under the modular automorphism group of the free quasi-free state is necessarily contained in the almost periodic free summand.
In this article, we proved the following results. Let $L(F(n_i))$ be the free group factor on $n_i$ generators and $λ(g_{i})$ be one of standard generators of $L(F(n_i))$ for $1\le i\le N$. Let $\A_i$ be the abelian von Neumann subalgebra of $L(F(n_i))$ generated by $λ(g_{i})$. Then the abelian von Neumann subalgebra $\otimes_{i=1}^N\A_i$ is a maximal injective von Neumann subalgebra of $\otimes_{i=1}^N L(F(n_i))$. When $N$ is equal to infinity, we obtained McDuff factors that contain maximal injective abelian von Neumann subalgebras.
We consider some conditions similar to Ozawa's condition (AO), and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a corollary, we prove that $\rm II_1$ factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type $\rm III_1$ factors with such a condition are semisolid as a $\rm II_\infty$ factor.
In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type $\tto$ factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity.
Let $M$ be a II$_1$ factor with a von Neumann subalgebra $Q\subset M$ that has infinite index under any projection in $Q'\cap M$ (e.g., $Q$ abelian; or $Q$ an irreducible subfactor with infinite Jones index). We prove that given any separable subalgebra $B$ of the ultrapower II$_1$ factor $M^ω$, for a non-principal ultrafilter $ω$ on $\Bbb N$, there exists a unitary element $u\in M^ω$ such that $uBu^*$ is orthogonal to $Q^ω$.
We use the free entropy defined by D. Voiculescu to prove that the free group factors can not be decomposed as closed linear spans of noncommutative monomials in elements of nonprime subfactors or abelian $*$-subalgebras, if the degrees of monomials have an upper bound depending on the number of generators. The resulting estimates for the hyperfinite and abelian dimensions of free group factors settle in the affirmative a conjecture of L. Ge and S. Popa (for infinitely many generators).
Let $M_n$ be a sequence of finite factors with $\dim(M_n)\rightarrow \infty$ and denote $\text{\bf M}=Π_ωM_n$ their ultraproduct over a free ultrafilter $ω$. We prove that if $\text{\bf Q}\subset \text{\bf M}$ is either an ultraproduct $\text{\bf Q}=Π_ωQ_n$ of subalgebras $Q_n\subset M_n$, with $Q_n \not\prec_{M_n} Q_n'\cap M_n$, $\forall n$, or the centralizer $\text{\bf Q}=B'\cap \text{\bf M}$ of a separable amenable *-subalgebra $B\subset \text{\bf M}$, then for any separable subspace $X\subset \text{\bf M}\ominus (\text{\bf Q}'\cap \text{\bf M})$, there exists a diffuse abelian von Neumann…
An $\widetilde A_2$ group $Γ$ acts simply transitively on the vertices of an affine building $\triangle$. We study certain subgroups $Γ_0 \cong {\Bbb Z}^2$ which act on certain apartments of $\triangle$. If one of these subgroups acts simply transitively on an apartment, then the corresponding subalgebra of the group von Neumann algebra is maximal abelian and singular. Moreover the Pukánszky invariant contains a type $I_{\infty}$ summand.
Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $σ$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenabi…
We prove a basic result about tensor products of a $\text{II}_1$ factor with a finite von Neumann algebra and use it to answer, affirmatively, a question asked by S. Popa about maximal injective factors.
We show that a free product of a II_1-factor and a finite von Neumann algebra with amalgamation over a finite dimensional subalgebra is always a II_1-factor, and provide an algorithm for describing it in terms of free products (with amalgamation over the scalars) and compression/dilation. As an application, we show that the class of direct sums of finitely many von Neumann algebras that are interpolated free group factors, hyperfinite II_1-factors, type I_n algebras for n finite, and finite dimensional algebras, is closed under taking free products with amalgamation over finite dimensional sub…
The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$ respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their $\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that for every ring isomorphism $Φ: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $Ψ: \cM \to \…
To every subfactor planar algebra was associated a II_1 factor with a canonical abelian subalgebra generated by the cup tangle. Using Popa's approximative orthogonality property, we show that this cup subalgebra is maximal amenable.
This paper studies weakly mixing (singular) and mixing masas in type $\rm{II}_{1}$ factors from a bimodule point of view. Several necessary and sufficient conditions to characterize the normalizing algebra of a masa are presented. We also study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures, arising from semidirect products, which ar…
In this paper, we give examples of maximal amenable subalgebras of the free group factor of two generators. More precisely, we consider two copies of the hyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take a Haar unitary $u_i$ which generates a Cartan subalgebra of it. We show that the von Neumann subalgebra generated by the self-adjoint operator $u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. This provides infinitely many non-unitary conjugate maximal amenable MASAs.
We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,μ)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.
In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the k-folded tensor product of the free group factor (on N>1 generators). A. Sinclair and R. Smith introduced the class of strongly singular maximal abelian von Neumann algebras (MASA) of type II_1 factors. One of their examples was the Laplacian subalgebra of the free group factor, known to be a singularMASA. Using the techniques presented by A. Sinclair and R.Smith, we show that the Laplacian subalgebra of the k-folded tensor product of the free group factor with itself is a strongly singular…
We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer--Ricard and Boutonnet--Houdayer--Raum.
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and d…
We show a close relationship between non-degenerate smooth commuting squares of $II_1$-factors with all inclusions of finite index and inclusions of subfactor planar algebras by showing that each leads to a construction of the other. One direction of this uses the Guionnet-Jones-Shlyakhtenko construction.
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.
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $Γ$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a free ergodic probability measure preserving action of $Γ$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability …
In this paper, we exhibit strongly singular maximal abelian subalgebras living inside certain k-folded tensors of von Neumann group factors. The two classes of groups under consideration are the free groups of rank greater than 2 and the free product of m>2 groups, all finite and having the same order p>2 (m>p-1). We prove actually more, namely that the unique trace preserving conditional expectations from these group factors onto their Laplacian subalgebras are asymptotic homomorphisms, which in turn implies that the subalgebras are strongly singular masas.
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ 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 $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
We construct a class of II_1 factors M that admit unclassifiably many Cartan subalgebras in the sense that the equivalence relation of being conjugate by an automorphism of M is complete analytic, in particular non Borel. We also construct a II_1 factor that admits uncountably many non isomorphic group measure space decompositions, all involving the same group G. So G is a group that admits uncountably many non stably orbit equivalent actions whose crossed product II_1 factors are all isomorphic.
We prove that the regular von Neumann subalgebras $B$ of the hyperfinite II_1 factor $R$ satisfying the condition $B'\cap R=Z(B)$ are completely classified (up to conjugacy by an automorphism of $R$) by the associated discrete measured groupoid $G$. We obtain a similar classification result for triple inclusions $A\subset B \subset R$, 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 of $G$ on $B$. We in fact prove two very general va…
We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra $\ell^\infty\Bbb N\subset \Cal B(\ell^2\Bbb N)$ have unique state extensions to $\Cal B(\ell^2\Bbb N)$, is equivalent to a similar statement in II$_1$ factor framework, concerning the ultrapower inclusion $D^ω\subset R^ω$, where $D$ is the Cartan subalgebra of the hyperfinite II$_1$ factor $R$, and $ω$ is a free ultraflter. While we do not settle the problem in this latter form, we prove that if $A$ is any singular maximal abelian subalgebra of $R$, then the inclusion $A^ω\subset R^ω$ does satisf…
We prove that the radial masa C in the free group factor is disjoint from other maximal amenable subalgebras in the following sense: any distinct maximal amenable subalgebra cannot have diffuse intersection with C.
We show that for each (0<λ<1), the free Araki-Woods factor of type III(_λ) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime), and does not contain a Cartan subalgebra.
Guionnet et al. gave a construction of a II_1 factor associated to a subfactor planar algebra. In this paper we define an unshaded planar algebra. To any unshaded planar algebra P we associate a finite von Neumann algebra M_P. We prove that M_P is a II_1 factor that contains a generic maximal abelian subalgebra called the cup subalgebra.
We find a description of the restriction of doubly stochastic maps to separable abelian $C^*$-subalgebras of a II$_1$ factor $\cM$. We use this local form of doubly stochastic maps to develop a notion of joint majorization between $n$-tuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II$_1$ factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian $C^*$-subalgebra of $\cM$ can be embedded into a separable abelia…
We give a new proof of a result of Ozawa showing that if a von Neumann subalgebra $Q$ of a free group factor $L\Bbb F_n, 2\leq n\leq \infty$ has relative commutant diffuse (i.e. without atoms), then $Q$ is amenable.
In this paper we prove that for a type II_1 factor N with a Cartan maximal abelian subalgebra (masa), the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.
In this paper we use the description of free group factors as the von Neumann algebras of Berezin's deformation of the upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $$ {\cal L}(x)=Λ(x)-(1/…
We establish a Galois correspondence for finite quantum groupoid actions on II_1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such a subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain C^*-algebras related to it.
In this paper we introduce and study strongly singular maximal abelian self-adjoint subalgebras of type $II_1$ factors. We show that certain elements of free groups and of non-elementary hyperbolic groups generate such masas, and these also give new examples of masas for which Popa's invariant delta is 1. We also explore the connection between Popa's invariant and strong singularity.
We prove that for any free ergodic nonsingular nonamenable action Γ \actson (X,μ) of all Γ in a large class of groups including all hyperbolic groups, the associated group measure space factor $L^\infty(X) \rtimes Γ$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in [PV12]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products $M_1 *_B M_2$ over a subalgebra B of type I.
For each 1<s<\infty, a Popa algebra A_s is constructed that embeds as a weakly dense C*-subalgebra of the interpolated free group factor L(F_s). Certain approximation properties for A_s are shown. It follows that L(F_s) has the weak expectation property of Lance with respect to A_s. In the course of the demonstration, it is proved that under certain conditions, full amalgamated free products of matrix algebras are residually finite dimensional.
We investigate maximal abelian subalgebras (masas) in separably acting type $II_1$ factors. We use the notion of distance between masas which we introduced in an earlier paper in this archive, OA/0107075. The main result of the paper is to show that for any unitary u and a masa A, the distance between A and uAu* is comparable to the distance in 2-norm from u to the unitary normalizer N(A) of A. In qualitative terms, this says that a unitary almost normalizes A if and only if it is close to a normalizing unitary. Inequalities in the paper make this statement quantitatively precise. As a consequ…
We show that any type ${\rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type ${\rm III_1}$ factors.
We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation $δ:M\rightarrow L^2(M)\bar{\otimes}L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $δ$ is moreover "algebraic" (i.e. its domain $M_0$ is finitely generated, $δ(M_0)\subset M_0\otimes M_0$ and $δ^*(1\otimes 1)\in M_0$), then we show that $M$ has no Cartan subalgebra. We also giv…
Richard Kadison showed that not every commutative von Neumann subalgebra of a factor von Neumann algebra is equal to its relative double commutant. We prove that every commutative C*-subalgebra of a centrally prime C*-algebra $B$ equals its relative approximate double commutant. If $B$ is a von Neumann algebra, there is a related distance formula.
