For any finite dimensional C*-algebra A with any trace vector {\vec s} whose components are rational numbers, we give an endomorphism Φ of the hyperfinite II_1 factor R such that: forall k in {\mathbb N} Φ^k (R)' \cap R= \otimes^k A The canonical trace τ on R extends the trace vector {\vec s} on A. As a corollary, we construct a one-parameter family of inclusions of hyperfinite II_1 factors N^λ \subset M^λ with trivial relative commutant (N^λ)' \cap M^λ= {\mathbb C} and with the Jones index [M^λ: N^λ]= λ^{-1} \in (4, \infty) \cap {\mathbb Q} This partially solves the problem of finding all possible values of indices of subfactors with trivial relative commutant in the hyperfinite II_1 factor, by showing that any rational number λ^{-1} > 4 can occur.
We introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by Kawahigashi-Sutherland-Takesaki in automorphism case.
Previously, we have shown that the CAR algebra for fermions is embedded in the Cuntz algebra ${\cal O}_{2}$ in such a way that the generators are expressed in terms of polynomials in the canonical generators of the latter, and it coincides with the U(1)-fixed point subalgebra ${\cal A}\equiv {\cal O}_{2}^{U(1)}$ of ${\cal O}_{2}$ for the canonical gauge action. Based on this embedding formula, some properties of ${\cal A}$ are studied in detail by restricting those of ${\cal O}_{2}$. Various endomorphisms of ${\cal O}_{2}$, which are defined by polynomials in the canonical generators, are expl…
Based on the structure of the hyperfinite $II_1$ factor, we study its Dirichlet forms which can be obtained from Dirichlet forms on $M_{2^n}(\mathbb{C})$
For any finite dimensional C^*-algebra A, we give an endomorphism Φof the hyperfinite II_1 factor R of finite Jones index such that: for all k \in \mathbb {N}, Φ^k (R)' \cap R= \otimes^k A. The Jones index [R: Φ(R)]= (rank (A))^2, here rank (A) is the dimension of the maximal abelian subalgebra of A.
In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is semifinite and $\N$ is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III$_0$-factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors $\M$ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III$_1$-factor. In contr…
W. Arveson has described a cocycle conjugacy class $U(α)$ of $E_0$-semigroup $α$ on B(H) which is a factor of type $\rm I$. Under some conditions on $α$, there is a $E_0$-semigroup $β\in U(α)$ being a flow of shifts in the sence of R.T. Powers. We study quasifree endomorphisms semigroups $α$ on the hyperfinite factor $M=π(A(K))''$ generated by the representations $π$ of the algebra of canonical anticommutation relations A(K) over a separable Hilbert space K. The type of M can be $\rm I$, $\rm II$ or $\rm III$ depending on $π$. The cocycle conjugacy class $U(α)$ is described in the terms of ini…
We investigate certain classes of normal completely positive (CP) maps on the hyperfinite $II_1$ factor $\mathcal A$. Using the representation theory of a suitable irrational rotation algebra, we propose some computable invariants for such CP maps.
For any finite dimensional C^*-algebra A with a trace vector \vec s whose entries are rational numbers, we give an endomorphism Φof the hyperfinite II_1 factor R such that: for all k \in \mathbb {N}, Φ^k (R)' \cap R= \otimes^k A. The canonical trace τon R extends the trace vector \vec s on A. Therefore the minimal projection is not necessarily equivalent to each other.
We construct numerous continuous families of irreducible subfactors of the hyperfinite II$_1$ factor, which are non-isomorphic, but have all the same standard invariant. In particular, we obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II$_1$ factor, which have Jones index 6 and have all the same standard invariant with property (T). We exploit the fact that property (T) groups have uncountably many non-cocycle conjugate cocycle actions on the hyperfinite II$_1$ factor.
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 begin this note with a von Neumann algebraic version of the elementary but extremely useful fact about being able to extend inner-product preserving maps from a total set of the domain Hilbert space to an isometry defined on the entire domain. This leads us to the notion of when `good' endomorphisms of a factorial probability space $(M,ϕ)$ (which we call equi-modular) admit a natural extension to endomorphisms of $L^2(M,ϕ)$. We exhibit examples of such extendable endomorphisms. We then pass to $E_0$-semigroups $α= {α_t: t \geq 0}$ of factors, and observe that extendability of this semigroup…
Using methods of R.J.Tauer we exhibit an uncountable family of singular masas in the hyperfinite $\textrm{II}_1$ factor $\R$ all with Pukánszky invariant $\{1\}$, no pair of which are conjugate by an automorphism of $R$. This is done by introducing an invariant $Γ(A)$ for a masa $A$ in a \IIi factor $N$ as the maximal size of a projection $e\in A$ for which $A e$ contains non-trivial centralising sequences for $eN e$. The masas produced give rise to a continuous map from the interval $[0,1]$ into the singular masas in $\R$ equiped with the $d_{\infty,2}$-metric. A result is also given showing …
Let A be a maximal abelian self-adjoint subalgebra (masa) in a type II_1 factor M acting via standard representation on L^2(M). The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection e_A from L^2(M) onto L^2(A). Then A'(1-e_A) decomposes into a direct sum of type I_n algebras for n in {1,2,...,\infty}, and those n's which occur in the direct sum form a set called the Pukanszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S\cup {\infty} whe…
Let $R$ be the hyperfinite ${\rm II}_1$ factor and let $u,v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2πiθ} uv$ for an irrational number $θ$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, …
We introduce a non-commutative Walsh system and prove that it forms a Schauder basis in the Lp-spaces (1 < p < \infty) associated with the hyperfinite III_λ-factors (0 < λ<= 1).
We show that it is relatively consistent with ZFC that there exists a hyperfinite type $\mathrm{II}_1$-factor of density character $\aleph_1$ which is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.
We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\hbox{II}_1$, $\hbox{II}_\infty$, $\hbox{III}_λ$, $0 < λ\leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space).
We consider various statements that characterize the hyperfinite II$_1$ factors amongst embeddable II$_1$ factors in the non-embeddable situation. In particular, we show that "generically" a II$_1$ factor has the Jung property (which states that every embedding of itself into its ultrapower is unitarily conjugate to the diagonal embedding) if and only if it is self-tracially stable (which says that every such embedding has an approximate lifting). We prove that the enforceable factor, should it exist, has these equivalent properties. Our techniques are model-theoretic in nature. We also show h…
We provide an alternative proof for the extreme amenability of the unitary group of the hyperfinite II${}_1$-factor von Neumann algebra, endowed with the strong operator topology.
We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg's QWEP Conjecture and Tsirelson's Problem.+
On page 43 in \cite{Po83} Sorin Popa asked whether the following property holds: \emph{If $ω$ is a free ultrafilter on $\mathbb N$ and $\mathcal R_1\subseteq \mathcal R$ is an irreducible inclusion of hyperfinite II$_1$ factors such that $\mathcal R'\cap \mathcal R^ω\subseteq \mathcal R^ω_1$ does it follows that $\mathcal R_1=\mathcal R$?} In this short note we provide an affirmative answer to this question.
For each sequence $\{c_n\}_n$ in $l_{1}(\N)$ we define an operator $A$ in the hyperfinite $\mathrm{II}_1$-factor $\mathcal{R}$. We prove that these operators are quasinilpotent and they generate the whole hyperfinite $\mathrm{II}_1$-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of $A$ are equally distributed, and we …
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $…
The free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$. The finite dimensional part depends on the minimal projections of the original algebras and the "dimension", r, of the free group factor part is found using the notion of free dimension. For discrete amenable groups $G$ and $H$ this implies that the group von Neumann algebra $L(G*H)$ is an interpolated free group factor and depends only on the orders of $G$ and $H$.
Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary self adjoint operators. (2) A McDuff factor embeds into the ultrapower of the hyperfinite II_1 factor if and only if it contains a weakly dense operator system which is injective. (3) There exists a simple, quasidiagonal, real rank zero C*-algebra with non-hyperfinite II_1 factor representations and which is not tracially AF. This answers negatively questio…
We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are given from the viewpoint of ergodic theory.
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 characterize the condition for two finite index endomorphisms on an AFD factor to be approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.
It is a well-known fact that endomorphisms of $B(H)$ are intimately connected with families of mutually orthogonal isometries, i.e. with representations of the so-called Toeplitz $C^*$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^*$-algebras associated to graphs and endomorphisms of certain von Neumann subalgebras of $B(H)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.
A II_1 factor embeds into the ultraproduct of the hyperfinite II_1 factor if and only if it satisfies the W*-analogue of Lance's weak expectation property. This note gives a self contained proof of this fact.
We construct a simple C*-algebra with nuclear dimension zero that is not isomorphic to its tensor product with the Jiang-Su algebra Z, and a hyperfinite II_1 factor not isomorphic to its tensor product with the separable hyperfinite II_1 factor R. The proofs use a weakening of the Continuum Hypothesis.
In 1965 Tauer produced a countably infinite family of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each $n\in\mathbb N$, finding masas for which this procedure terminates at the $n$-th stage. Such masas are said to have length $n$. In this paper we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the…
We show that every binary shift on the hyperfinite $II_1$ factor $R$ is cocycle conjugate to at least countably many non-conjugate binary shifts. This holds in particular for binary shifts of infinite commutant index.
In \cite{Ioana:vNsuperrigidity}, Ioana introduced three new invariants of type II$_1$ factors: the one-sided fundamental group, the endomorphism semigroup and the set of right-finite bimodules. In \cite{Ioana:vNsuperrigidity}, he does not provide many computations of these invariants. In particular, the question whether these invariants can be trivial is left open. We give an explicit example of a type II$_1$ factor for which all three invariants are trivial. More generally, for any countable left-cancellative semigroup $G$, we construct a type II$_1$ factor $M$ whose endomorphism semigroup is…
We prove that any separable II$_1$ factor $M$ admits a {\it coarse decomposition} over the hyperfinite II$_1$ factor $R$, i.e., there exists an embedding $R\hookrightarrow M$ such that $L^2M\ominus L^2R$ is a multiple of the coarse Hilbert $R$-bimodule $L^2R \overline{\otimes} L^2R^{op}$ (equivalently, the von Neumann algebra generated by left and right multiplication by $R$ on $L^2M\ominus L^2R$ is isomorphic to $R\overline{\otimes}R^{op}$). Moreover, if $Q\subset M$ is an infinite index irreducible subfactor, then $R\hookrightarrow M$ can be constructed so that to also be coarse with respect…
We prove that any free cocycle action of a countable amenable group $Γ$ on any II$_1$ factor $N$ can be perturbed by inner automorphisms to a genuine action. This {\em vanishing cohomology} property, that we call $\mathcal V\mathcal C$, is also closed to free products with amalgamation over finite groups. But beyond this no other examples of $\mathcal V\mathcal C$-groups are known. In turn, by considering special cocycle actions $Γ\curvearrowright N$ in the case $N$ is the hyperfinite II$_1$ factor $R$, respectively the free group factor $N=L(\Bbb F_\infty)$, we exclude many groups from being …
We compute the index of the inclusions of type $III_{1/2}$ factors arising from endomorphisms of the Cuntz algebra ${\mathcal O}_2$ associated to the rank-two permutation matrices.
We determine the subfactors $N\subset R$ of the hyperfinite $II_1$-factor R with finite index for which the $C^*$-tensor category of the associated $(N,N)$-bimodules is equivalent to the $C^*$-tensor category $\C{U}_G$ of all unitary finite dimensional representations of a given finite group G.
We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group $B_0^{\mathbb N}$. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type ${\rm I}_\infty$ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,1…
We study the von Neumann algebra, generated by the regular representations of the infinite-dimensional nilpotent group $B_0^{\mathbb Z}$. In [14] a condition have been found on the measure for the right von Neumann algebra to be the commutant of the left one. In the present article, we prove that, in this case, the von Neumann algebra generated by the regular representations of group $B_0^{\mathbb Z}$ is the type ${\rm III}_1$ hyperfinite factor. We use a technique, developed in [20] where a similar result was proved for the group $B_0^{\mathbb N}$. The crossed product allows us to remove some…
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 …
