Inclusions Of Von Neumann Algebras Research Articles

A Dixmier type averaging property of automorphisms on a C∗-algebra

In this study of the relative Dixmier property for inclusions of von Neumann algebras and of [Formula: see text]-algebras, Popa considered a certain property of automorphisms on [Formula: see text]-algebras, that we here call the strong averaging property. In this paper, we characterize when an automorphism on a [Formula: see text]-algebra has the strong averaging property. In particular, automorphisms on commutative [Formula: see text]-algebras possess this property precisely when they are free. An automorphism on a unital separable simple [Formula: see text]-algebra with at least one tracial state has the strong averaging property precisely when its extension to the finite part of the bi-dual of the [Formula: see text]-algebra is properly outer, and in the simple, non-tracial case the strong averaging property is equivalent to being outer. To illustrate the usefulness of the strong averaging property we give three examples where we can provide simpler proofs of existing results on crossed product [Formula: see text]-algebras, and we are also able to extend these results in different directions.

Asymptotic Analysis of von Neumann Entropy in Conformal Field Theory

Given a QFT net {{mathcal {A}}} of local von Neumann algebras {{mathcal {A}}}(O), we consider the von Neumann entropy S_{{mathcal {A}}}(O, {widetilde{O}}) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras {{mathcal {A}}}(O)subset {{mathcal {A}}}({widetilde{O}}) (split property). This canonical entanglement entropy S_{{mathcal {A}}}(O, {widetilde{O}}) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). The finiteness property is derived by an explicit formula of entropy and an observation that the operators in the definition are closely related to Hankel operators. In this paper we give further analysis of this entropy using a variety of techniques that have been developed in different context, and in particular we show that there is an upper bound given by a positive constant multiply by |ln eta |, where eta is the cross ratio of the underlying system, when eta rightarrow 0.

Relative entropy and curved spacetimes

Given any half-sided modular inclusion of standard subspaces, we show that the entropy function associated with the decreasing one-parameter family of translated standard subspaces is convex for any given (not necessarily smooth) vector in the underlying Hilbert space. In second quantisation, this infers the convexity of the vacuum relative entropy with respect to the translation parameter of the modular tunnel of von Neumann algebras. This result allows us to study the QNEC inequality for coherent states in a free Quantum Field Theory on a stationary curved spacetime, given a KMS state. To this end, we define wedge regions and appropriate (deformed) subregions. Examples are given by the Schwarzschild spacetime and null translated subregions with respect to the time translation Killing flow. More generally, we define wedge and strip regions on a globally hyperbolic spacetime, so to have non trivial modular inclusions of von Neumann algebras, and make our analysis in this context.

On the weak relative Dixmier property

We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa \cite{Po99}. The proof uses an improvement of Ellis' lemma for compact convex semigroup.

Von Neumann Entropy in QFT

In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones $O \subset \tilde O$ of the spacetime, where the closure of $O$ is contained in $\tilde O$. Given a QFT net $A$ of local von Neumann algebras $A(O)$, we consider the von Neumann entropy $S_A(O, \tilde O)$ of the restriction of the vacuum state to the canonical intermediate type $I$ factor for the inclusion of von Neumann algebras $A(O)\subset A(\tilde O)$ (split property). We show that this canonical entanglement entropy $S_A(O, \tilde O)$ is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy $\underline S_A(O, \tilde O)$, the infimum of the vacuum von Neumann entropy of $F$, where $F$ here runs over all the intermediate, discrete type $I$ von Neumann algebras. We prove that $\underline S_A(O, \tilde O)$ is finite for the local chiral conformal net generated by finitely many commuting $U(1)$-currents.

Structure of bicentralizer algebras and inclusions of type $$\mathrm{III}$$ factors

We investigate the structure of the relative bicentralizer algebra $$\mathrm{B}(N \subset M, \varphi )$$ for inclusions of von Neumann algebras with normal expectation where N is a type $$\mathrm{III}_1$$ subfactor and $$\varphi \in N_*$$ is a faithful state. We first construct a canonical flow $$\beta ^\varphi : \mathbf {R}^*_+ \curvearrowright \mathrm{B}(N \subset M, \varphi )$$ on the relative bicentralizer algebra and we show that the W$$^*$$-dynamical system $$(\mathrm{B}(N \subset M, \varphi ), \beta ^\varphi )$$ is independent of the choice of $$\varphi $$ up to a canonical isomorphism. In the case when $$N=M$$, we deduce new results on the structure of the automorphism group of $$\mathrm{B}(M,\varphi )$$ and we relate the period of the flow $$\beta ^\varphi $$ to the tensorial absorption of Powers factors. For general irreducible inclusions $$N \subset M$$, we relate the ergodicity of the flow $$\beta ^\varphi $$ to the existence of irreducible AFD subfactors in M that sit with normal expectation in N. When the inclusion $$N \subset M$$ is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison’s problem when N is amenable.

From a Kac algebra subfactor to Drinfeld double

Given a finite-index and finite-depth subfactor, we define the notion of quantum double inclusion—a certain unital inclusion of von Neumann algebras constructed from the given subfactor—which is closely related to that of Ocneanu’s asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor RH ⊂ R produces Drinfeld double of H, where H is a finite-dimensional Kac algebra acting outerly on the hyperfinite II1 factor R and RH denotes the fixed-point subalgebra. More precisely, quantum double inclusion of RH ⊂ R is isomorphic to R ⊂ R ⋊ D(H)cop for some outer action of D(H)cop on R, where D(H) denotes the Drinfeld double of H.

On intermediate subalgebras of inclusions of von Neumann algebras having common Cartan subalgebras and their basic extensions

On intermediate subalgebras of inclusions of von Neumann algebras having common Cartan subalgebras and their basic extensions

Minimal Index and Dimension for 2-C*-Categories with Finite-Dimensional Centers

In the first part of this paper, we give a new look at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones' index. The minimal conditional expectation is characterized by means of a canonical state on the relative commutant, that we call the spherical state; the minimal index is neither additive nor multiplicative (it is submultiplicative), contrary to the subfactor case. So we introduce a matrix dimension with the good functorial properties: it is always additive and multiplicative. The minimal index turns out to be the square of the norm of the matrix dimension, as was known in the multi-matrix inclusion case. In the second part, we show how our results are valid in a purely 2-$C^*$-categorical context, in particular they can be formulated in the framework of Connes' bimodules over von Neumann algebras.

Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators f↦w⋅(f∘φ) on the Hardy space H2 on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).

Cohomology and $L^2$-Betti Numbers for Subfactors and Quasi-Regular Inclusions

Author(s): Popa, Sorin; Shlyakhtenko, Dimitri; Vaes, Stefaan | Abstract: We introduce $L^2$-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 II_1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups $\Gamma$, we recover the ordinary (co)homology of $\Gamma$. For Cartan subalgebras, we recover Gaboriau's $L^2$-Betti numbers for the associated equivalence relation. In this common framework, we prove that the $L^2$-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the $L^2$-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.

Weak mixing for locally compact quantum groups

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

Free independence in ultraproduct von Neumann algebras and applications

The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^\omega, \varphi^\omega)$ where $(M, \varphi)$ is a $\sigma$-finite von Neumann algebra endowed with a faithful normal state satisfying $(M^\varphi)' \cap M = \mathbf{C} 1$. More precisely, we show that whenever $P_1, P_2 \subset M^\omega$ are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group $(\sigma_t^{\varphi^\omega})$, there exists a unitary $v \in \mathcal U((M^\omega)^{\varphi^\omega})$ such that $P_1$ and $v P_2 v^*$ are $\ast$-free inside $M^\omega$ with respect to the ultraproduct state $\varphi^\omega$. Combining our main result with the recent work of Ando-Haagerup-Winsl\o w [AHW13], we obtain a new and direct proof, without relying on Connes-Tomita-Takesaki modular theory, that Kirchberg's quotient weak expectation property (QWEP) for von Neumann algebras is stable under free product. Finally, we obtain a new class of inclusions of von Neumann algebras with the relative Dixmier property.

Classification of actions of compact abelian groups on subfactors with index less than 4

We classify actions of discrete abelian groups on some inclusions of von Neumann algebras, up to cocycle conjugacy. As an application, we classify actions of compact abelian groups on the inclusions of approximately finite dimensional (AFD) factors of type II1 with index less than 4, up to stable conjugacy.

Outer actions of measured quantum groupoids

Mimicking a recent article of Stefaan Vaes, in which was proved that every locally compact quantum group can act outerly, we prove that we get the same result for measured quantum groupoids, with an appropriate definition of outer actions of measured quantum groupoids. This result is used to show that every measured quantum groupoid can be found from some depth 2 inclusion of von Neumann algebras.

Relative matched pairs of finite groups from depth two inclusions of von Neumann algebras to quantum groupoids

In this work we give a generalization of matched pairs of (finite) groups to describe a general class of depth two inclusions of factor von Neumann algebras and the C ∗ -quantum groupoids associated with, using double groupoids.

Endomorphisms on half-sided modular inclusions

In algebraic quantum field theory we consider nets of von Neumann algebras indexed over regions of the space time. Wiesbrock [“Conformal quantum field theory and half-sided modular inclusions of von Neumann algebras,” Commun. Math. Phys. 158, 537–543 (1993)] has shown that strongly additive nets of von Neumann algebras on the circle are in correspondence with standard half-sided modular inclusions. We show that a finite index endomorphism on a half-sided modular inclusion extends to a finite index endomorphism on the corresponding net of von Neumann algebras on the circle. Moreover, we present another approach to encoding endomorphisms on nets of von Neumann algebras on the circle into half-sided modular inclusions. There is a natural way to associate a weight to a Möbius covariant endomorphism. The properties of this weight have been studied by Bertozzini et al. [“Covariant sectors with infinite dimension and positivity of the energy,” Commun. Math. Phys. 193, 471–492 (1998)]. In this paper we show the converse, namely, how to associate a Möbius covariant endomorphism to a given weight under certain assumptions, thus obtaining a correspondence between a class of weights on a half-sided modular inclusion and a subclass of the Möbius covariant endomorphisms on the associated net of von Neumann algebras. This allows us to treat Möbius covariant endomorphisms in terms of weights on half-sided modular inclusions. As our aim is to provide a framework for treating endomorphisms on nets of von Neumann algebras in terms of the apparently simpler objects of weights on half-sided modular inclusions, we lastly give some basic results for manipulations with such weights.

Inclusions of von Neumann algebras and quantum groupoïds III

In a former article, in collaboration with Jean–Michel Vallin, we have constructed two “quantum groupoı¨ds” dual to each other, from a depth 2 inclusion of von Neumann algebras M 0 ⊂ M 1 , in such a way that the canonical Jones’tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures. We are now investigating in greater details these structures in the presence of an appropriate modular theory on the basis M 0 ′ ∩ M 1 , and we show how these examples fit with Lesieur's “measured quantum groupoı¨ds”.

QUANTUM GROUPOIDS OF COMPACT TYPE

To any groupoid, equipped with a Haar system, Jean-Michel Vallin had associated several objects (pseudo-multiplicative unitary, Hopf-bimodule) in order to generalize, up to the groupoid case, the classical notions of multiplicative unitary and Hopf–von Neumann algebra, which were intensely used to construct quantum groups in the operator algebra setting. In two former articles (one in collaboration with Jean-Michel Vallin), starting from a depth-2 inclusion of von Neumann algebras, we have constructed such objects, which allowed us to study two ‘quantum groupoids’ dual to each other. We are now investigating in greater details the notion of pseudo-multiplicative unitary, following the general strategy developed by Baaj and Skandalis for multiplicative unitaries. AMS 2000 Mathematics subject classification: Primary 46L89; 22A22; 81R50

Correspondences and Approximation Properties for von Neumann Algebras

The notion of the amenability of a locally compact group has been extended in various ways. Two weaker versions of amenability, weak amenability and the approximation property, have been defined for locally compact groups (by Haagerup and Haagerup and Kraus, respectively) and Bekka has defined a notion of amenability for representations of locally compact groups. Correspondences can be viewed as a generalization of representations of such groups. Using this viewpoint, Ananthraman–Delaroche has defined a notion of (left) amenability for correspondences. In this paper, we define notions of weak amenability and the approximation property for correspondences (and representations of locally compact groups), and obtain various results concerning these notions. Ananthraman–Delaroche showed that if N ⊂ M is an inclusion of von Neumann algebras, and if the associated inclusion correspondence is left amenable, then various approximation properties of N (semidiscreteness, the weak* completely bounded approximation property, and the weak* operator approximation property) are shared by M. We show that if this correspondence has the (weaker) approximation property, then if N has the weak* operator approximation property, so does M. An application of this result to crossed products is also given.