Haagerup Approximation Property Research Articles

Mixed $q$-deformed Araki–Woods von Neumann algebras

Given a strongly continuous orthogonal representation (U_t)_{t\in \mathbb{R}} of \mathbb{R} on a real Hilbert space {\mathcal{H}}_{\mathbb{R}} , a decomposition {\mathcal{H}}_{\mathbb{R}}:=\bigoplus_{i\in N}\mathcal{H}_{\mathbb{R}}^{(i)} consisting of invariant subspaces of (U_t)_{t\in\mathbb{R}} and an appropriate matrix ((q_{ij}))_{N\times N} of real parameters, we associate representations of the mixed commutation relations on twisted Fock spaces. The associated von Neumann algebras are (usually) non-tracial and are generalizations of those constructed by Bożejko–Speicher and Hiai. We investigate the factoriality of these von Neumann algebras. Along the process, we show that the generating abelian subalgebras associated with the blocks of the aforesaid decomposition are strongly mixing masas when they admit appropriate conditional expectations. On the contrary, the generating abelian algebras which fail to admit appropriate conditional expectations are quasi-split. We also discuss non-injectivity and the Haagerup approximation property.

Relative Haagerup property for arbitrary von Neumann algebras

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

The Haagerup approximation property for arbitrary C*-algebras

In this paper, we generalize the notion of the Haagerup approximation property to arbitrary C-algebras. We also prove the permanence of the Haagerup approximation property under a few canonical constructions of C-algebras.

Haagerup approximation property via bimodules

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

$q$-Araki-Woods algebras: Extension of second quantisation and Haagerup approximation property

We extend the class of contractions for which the second quantisation on $q$-Araki-Woods algebras can be defined. As a corollary, we prove that all $q$-Araki-Woods algebras possess the Haagerup approximation property.

Haagerup approximation property and positive cones associated with a von Neumann algebra

We introduce the notion of the $\alpha$-Haagerup approximation property for $\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $\alpha$-Haagerup approximation property actually does not depend on a choice of $\alpha$. This gives us a direct proof of the fact that two characterizations of the Haagerup approximation property are equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the $L^p$-Haagerup approximation property for a non-commutative $L^p$-spaces associated with a von Neumann algebra ($1<p<\infty$) and show the independency of the $L^p$-Haagerup approximation property on $p$.

Haagerup Approximation Property for Arbitrary von Neumann Algebras

We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.

Compact operators and algebraic K-theory for groups which act properly and isometrically on Hilbert space

Abstract We prove the K-theoretic Farrell–Jones conjecture for groups with the Haagerup approximation property and coefficient rings and C * C^{*} -algebras which are stable with respect to compact operators. We use this and Higson–Kasparov’s result that the Baum–Connes conjecture holds for such a group G, to show that the algebraic and the C * C^{*} -crossed product of G with a stable separable G- C * C^{*} -algebra have the same K-theory.

The Haagerup Approximation Property for von Neumann Algebras via Quantum Markov Semigroups and Dirichlet Forms

The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state φ is shown to imply existence of unital, φ-preserving and KMS-symmetric approximating maps. This is used to obtain a characterisation of the Haagerup approximation property via quantum Markov semigroups (extending the tracial case result due to Jolissaint and Martin) and further via quantum Dirichlet forms.

A Cocycle in the Adjoint Representation of the Orthogonal Free Quantum Groups

We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that the associated von Neumann algebras are full II1-factors and Brannan's result showing that the orthogonal free quantum groups have Haagerup's approximation property. We also deduce Ozawa-Popa's property strong (HH) and give a new proof of Isono's result about strong solidity.

The fusion rules of some free wreath product quantum groups and applications

In this article we find the fusion rules of the free wreath product quantum groups Γˆ≀⁎SN+ for any discrete group Γ. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced C⁎-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type II1-factor and that the dual of Γˆ≀⁎SN+ has the Haagerup approximation property for all finite groups Γ.

Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.

Approximation properties for free orthogonal and free unitary quantum groups

We show that the reduced von Neumann algebras of the free orthogonal and free unitary quantum groups have the Haagerup approximation property. Using this result and a Haagerup-type inequality for these quantum groups due to Vergnioux (J. Operator Theory 57, 303-324 (2007)), we also show that the reduced C$^\ast$-algebras of the free orthogonal and free unitary quantum groups have the metric approximation property.

Mixing actions of groups with the Haagerup approximation property

Mixing actions of groups with the Haagerup approximation property

Borel cocycles, approximation properties and relative property T

Let $G$ and $H$ be locally compact groups. Assume that $G$ acts on a standard probability space $(S,\mu)$, $\mu$ being $G$-invariant. We prove that if there exists a Borel cocycle $\alpha:S\times G\longrightarrow H$ which is proper in an appropriate sense, then $G$ inherits some approximation properties of $H$, for instance amenability or the so-called Haagerup Approximation Property. On the other hand, if $G_{0}$ is a closed subgroup of $G$, if the pair $(G,G_{0})$ has the relative property (T) of Margulis [19] and if either $H$ has Haagerup Approximation Property, or if it is the unitary group of a finite von Neumann algebra with a similar property, then we give rigidity results analogous to that in [23] and [1].

Compressions of free products of von Neumann algebras

A reduction formula for compressions of von Neumann algebra II $_1$ –factors arising as free products is proved. This shows that the fundamental group is ${\bf R}^*_+$ for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II $_\infty$ –factors are constructed; this produces type III $_1$ factors with core $\mathcal{M}\otimes B(\mathcal{H})$ , where $\mathcal{M}$ can be a full II $_1$ –factor without the Haagerup approximation property.

Positive definite functions and relative property (T) for subgroups of discrete groups

Relative Property (T) for a subgroup H of a group G and its connection with positive definite functions are studied. A relation with the Haagerup approximation property is established. We show that if H is a non-normal subgroup of a group G which has Property (T) and G/H is amenable as a graph then H has finite index in G.

On the method of constructing irreducible finite index subfactors of Popa

Let US(Q) be the universal Jones algebra associated to a finite von Neumann algebra Q and Rs c R be the Jones subfactors, s € {4cos2 \\n > 3} [4, oo). We consider for any von Neumann subalgebra Qo C Q the algebra US(Q, Qo) defined as the quotient of US(Q) through its ideal generated by [Q o, R] and we construct a Markov trace on US(Q, Qo). If Z(Q) Π Z(Q*) = C and Q contains n > s + I unitaries U = 1, Uι, ... , un, with EQo(u*Uj) = δijl , 1 < /, j' < n, then we get a family of irreducible inclusions of type Hi factors Ns c Ms, with [Ms : Ns] = 5 and minimal higher relative commutant. Although these subfactors are nonhyperfinite, they have the Haagerup approximation property whether Qo C Q is a Haagerup inclusion and if either Qo is finite dimensional or Qo C