Product Von Neumann Algebras Research Articles

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.

A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

We obtain a Mikhlin multiplier theory for the nonabelian free groups. Let F∞ be a free group on infinite many generators {g1,g2,⋯}. Given d≥1 and a bounded symbol m on Zd satisfying the classical Mikhlin condition, the linear map Mm:C[F∞]→C[F∞] defined by λ(g)↦m(k1,⋯,kd)λ(g) for g=gi1k1⋯ginkn∈F∞ in reduced form (with kl=0 in m(k1,⋯,kd) for l>n), extends to a completely bounded map on Lp(Fˆ∞) for all 1<p<∞, where Fˆ∞ is the group von Neumann algebra of F∞. In the process, we establish a platform to transfer Lp-completely bounded maps on tensor products of von Neumann algebras to Lp-completely bounded maps on the corresponding amalgamated free products. A similar result holds for any free product of discrete groups.

Matrix models for varvec{varepsilon }-free independence

We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to varepsilon -free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular varepsilon arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary varepsilon may be realized in this way. As a result, we obtain a new proof that mathcal {R}^omega -embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.

E0-semigroups of free Araki-Woods factors

Analyzing endomorphisms and shift E0-semigroups of free products of von Neumann algebras, we construct uncountably many pairwise cocycle inequivalent E0-semigroups on a large class of free Araki-Woods factors. These E0-semigroups are equimodular but not modularly extendable. We also produce conjugate but not cocycle equivalent E0-semigroups on non-type I factors.

Free Hilbert transforms

We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This implies that the decomposition of the free group $\F_\infty$ into reduced words starting with distinct free generators is completely unconditional in $L^p$. We study the case of Voiculescu's amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness-problem posed by Ozawa, a length independent estimate for Junge-Parcet-Xu's free Rosenthal inequality, a Littlewood-Paley-Stein type inequality for geodesic paths of free groups, and a length reduction formula for $L^p$-norms of free group von Neumann algebras.

Bimodules in Group Graded Rings

In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = ⊕g∈GRg be a unital G-graded ring. We say that R is G-controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) Re-sub-bimodules of R, given by G ⊇ H↦ ⊕h∈HRh. For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with Re ⊆ T ⊆ R of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.

Intermediate subalgebras and bimodules for general crossed products of von Neumann algebras

Let [Formula: see text] be a discrete group acting on a von Neumann algebra [Formula: see text] by properly outer ∗-automorphisms. In this paper, we study the containment [Formula: see text] of [Formula: see text] inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the [Formula: see text]-bimodules that are closed in the Bures topology and which coincide with the [Formula: see text]-closed ones under a mild hypothesis on [Formula: see text]. We use these results to obtain a general version of Mercer’s theorem concerning the extension of certain isometric [Formula: see text]-continuous maps on [Formula: see text]-bimodules to ∗-automorphisms of the containing von Neumann algebras.

Random and Free Positive Maps with Applications to Entanglement Detection

We apply random matrix and free probability techniques to the study of linear maps of interest in quantum information theory. Random quantum channels have already been widely investigated with spectacular success. Here, we are interested in more general maps, asking only for $k$-positivity instead of the complete positivity required of quantum channels. Unlike the theory of completely positive maps, the theory of $k$-positive maps is far from being completely understood, and our techniques give many new parametrized families of such maps. We also establish a conceptual link with free probability theory, and show that our constructions can be obtained to some extent without random techniques in the setup of free products of von Neumann algebras. Finally, we study the properties of our examples and show that for some parameters, they are indecomposable. In particular, they can be used to detect the presence of entanglement missed by the partial transposition test, that is, PPT entanglement. As an application, we considerably refine our understanding of PPT states in the case where one of the spaces is large whereas the other one remains small.

Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M in the context of an inclusion M⊆M⋊αG, where G is a discrete group acting on a factor M by outer ⁎-automorphisms. We characterize the M-bimodules X⊆M⋊αG that are closed in the Bures topology in terms of the subsets of G. We show that this characterization also holds for w⁎-closed bimodules when G has the approximation property (AP), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w⁎-continuous surjective isometric maps on X.

Conformal Nets I: Coordinate-Free Nets

We describe a coordinate-free perspective on conformal nets, as functors from intervals to von Neumann algebras. We discuss an operation of fusion of intervals and observe that a conformal net takes a fused interval to the fiber product of von Neumann algebras. Though coordinate-free nets do not a priori have vacuum sectors, we show that there is a vacuum sector canonically associated to any circle equipped with a conformal structure. This is the first in a series of papers constructing a 3-category of conformal nets, defects, sectors, and intertwiners.

Similarity degrees for the crossed product of von Neumann algebras

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.

When do pieces determine the whole? Extreme marginals of a completely positive map

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.

Some analysis of amalgamated free products of von Neumann algebras in the non-tracial setting

Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.

On Some Free Products of Von Neumann Algebras which are Free Araki-Woods Factors

We prove that certain free products of factors of type I and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki–Woods factors. In particular, they have the free absorption property and Connes' Sd invariant completely classifies these free products. For example, for λ, μ ∈]0, 1[, we show that(M2(C),ωλ)*(M2(C),ωμ)is isomorphic to the free Araki–Woods factor whose Sd invariant is the subgroup of R+* generated by λ and μ. Our proofs are based on algebraic techniques and amalgamated free products. These results give some answers to questions of Dykema and Shlyakhtenko.

Bass-Serre rigidity results in von Neumann algebras

We obtain new Bass-Serre type rigidity results for ${\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras $\mathcal{M}_1 \ast_B \mathcal{M}_2$ over an abelian von Neumann algebra $B$, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\rm II_1}$ and in the type ${\rm III}$ case, new examples of prime factors.

The splitting problem for subspaces of tensor products of operator algebras

The main result of this paper is that if N N is a von Neumann algebra that is a factor and has the weak* operator approximation property (the weak* OAP), and if R R is a von Neumann algebra, then every σ \sigma -weakly closed subspace of N ⊗ ¯ R {{N}\bar \otimes {R}} that is an N ⊗ ¯ C 1 R {N}\bar \otimes {\mathbb {C} 1_{R}} -bimodule (under multiplication) splits, in the sense that there is a σ \sigma -weakly closed subspace T T of R R such that S = N ⊗ ¯ T S={{N}\bar \otimes {T}} . Note that if S S is a von Neumann subalgebra of N ⊗ ¯ R {{N}\bar \otimes {R}} , then S S is an N ⊗ ¯ C 1 R {N}\bar \otimes {\mathbb {C} 1_{R}} -bimodule if and only if N ⊗ ¯ C 1 R ⊂ S {N}\bar \otimes {\mathbb {C} 1_{R}} \subset S . So this result is a generalization (in the case where N N has the weak* OAP) of the result of Ge and Kadison that if N N is a factor, then every von Neumann subalgebra M M of N ⊗ ¯ R {{N}\bar \otimes {R}} that contains N ⊗ ¯ C 1 R {N}\bar \otimes {\mathbb {C} 1_{R}} splits. We also obtain other results concerning the splitting of σ \sigma -weakly closed subspaces of tensor products of von Neumann algebras and the splitting of normed closed subspaces of C*-algebras that generalize results previously obtained for von Neumann subalgebras and C*-subalgebras.

Embeddings of reduced free products of operator algebras

Given reduced amalgamated free products of C$^*$--algebras $(A,\phi)=\freeprodi(A_\iota,\phi_\iota)$ and $(D,\psi)=\freeprodi(D_\iota,\psi_\iota)$, an embedding $A\hookrightarrow D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_\iota\hookrightarrow D_\iota$. This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, the reduced amalgamated free product of von Neumann algebras is defined in the general case and analogues of the above mentioned results are proved for von Neumann algebras.

Remarks on free products with respect to non-tracial states

Since the appearance of D. Voiculescu's free probability theory detailed study on free products of von Neumann algebras (based on his theory) has been carried by several authors, and free group factors are their typical examples. Indeed, his theory gives us very powerful tools to analyze free group factors. On the other hand, at the beginning of the 80's (i.e., before the appearance of D. Voiculescu's theory) S. Popa studied in detail free group factors and related type II1 factors and got several deep results ([P1], [P2]). He used there some techniques for type II1 factors related to his notion ``orthogonal pairs''. Then, at the mid 90's L. Ge generalized them to more general (but type II1) setting and solved some problems on maximal injective subalgebras ([G]). In this note, we will try to generalize the techniques to more general and not necessary type II1 setting. These generalizations are rather natural, but enable us to obtain some results on free products with respect to non-tracial states. In fact, as applications a factoriality and type classification result for free products and a result on (normal) conditional expectations from free products onto their subalgebras will be obtained. Based on the latter result we will discuss the recent question posed by M. Izumi, R. Longo and S. Popa ([ILP]): Let M L N be von Neumann algebras with a normal conditional expectation from M onto N. If N 0 \M ˆ C1, does there exist a normal conditional expectation from M onto L ?

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.

A-Valued Semicircular Systems

To a von Neurnann algebra A and a set of linear maps ηij:A→A, i, j∈I such that a↦(ηij)ij∈I:A→A⊗B(l2(I)) is normal and completely positive, we associate a von Neumann algebra Φ(A, η). This von Neumann algebra is generated by A and an A-valued semicircular system Xi, i∈I, associated to η. In many cases there is a faithful conditional expectation E:Φ(A, η)→A; if A is tracial, then under certain assumptions on η, Φ(A, η) also has a trace. One can think of the construction Φ(A, η) as an analogue of a crossed product construction. We show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction Φ. Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form Φ(L∞[0, 1], η). The cores of free Araki–Woods factors are free Krieger algebras. We study the free Krieger algebras and as a result obtain several non-isomorphism results for free Araki–Woods factors. As another source of classification results for free Araki–Woods factors, we compute the τ invariant of Connes for free products of von Neumann algebras. This computation generalizes earlier work on computation of T, S, and Sd invariants for free product algebras.