Free Product Von Neumann Algebras Research Articles

Maximal amenability with asymptotic orthogonality in amalgamated free products

We investigate the use of Popa's asymptotic orthogonality to establish maximal amenability for amalgamated free product von Neumann algebras. Although new techniques have recently been developed to consider amalgamated free products, we find that the technique developed by Popa in the 1980s can be used to demonstrate maximal amenability for a certain family of amalgamated free product von Neumann algebras.

Classification of a family of non-almost-periodic free Araki–Woods factors

Author(s): Houdayer, Cyril; Shlyakhtenko, Dimitri; Vaes, Stefaan | Abstract: We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors $\Gamma(\mu,m)$ up to isomorphism. We do this by showing that free Araki-Woods factors $\Gamma(\mu, m)$ arising from finite symmetric Borel measures $\mu$ on $\mathbf{R}$ whose atomic part $\mu_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})$ as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.

Free products in AQFT

We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take Mobius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with large relative commutant, by considering either a finite family of identical inclusions or an infinite family of inequivalent inclusions. In two dimensional spacetime, we construct Borchers triples with trivial relative commutant by taking free products of infinitely many, identical Borchers triples. Free products of finitely many Borchers triples are possibly associated with Haag-Kastler net having S-matrix which is nontrivial and non asymptotically complete, yet the nontriviality of double cone algebras remains open.

Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

AbstractWe investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.

Amenable absorption in amalgamated free product von Neumann algebras

We investigate the position of amenable subalgebras in arbitrary amalga- mated free product von Neumann algebras M=M1*BM2. Our main result states that, under natural analytic assumptions, any amenable subalgebra of M that has a large intersection with M1 is actually contained in M1. The proof does not rely on Popa’s asymptotic orthogonality property but on the study of nonnormal conditional expectations.

Rigidity of free product von Neumann algebras

Let $I$ be any nonempty set and let $(M_{i},\unicode[STIX]{x1D711}_{i})_{i\in I}$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class ${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$ of (possibly type $\text{III}$) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product $(M,\unicode[STIX]{x1D711})=\ast _{i\in I}(M_{i},\unicode[STIX]{x1D711}_{i})$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_{i}$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type $\text{II}_{1}$ factors and is new for free product type $\text{III}$ factors. It moreover provides new rigidity phenomena for type $\text{III}$ factors.

Asymptotic structure of free product von Neumann algebras

AbstractLet (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

Discrete cores of type III free product factors

We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.

A remark on amenable von Neumann subalgebras in a tracial free product

We give a simple proof of a theorem of C. Houdayer that an amenable von Neumann subalgebra in a tracial free product von Neumann algebra $M=M_{1}*M_{2}$ is contained in $M_{1}$ whenever it has a diffuse intersection with $M_{1}$.

Gamma Stability in Free Product von Neumann Algebras

Let \({(M, \varphi) = (M_1, \varphi_1) * (M_2, \varphi_2)}\) be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer \({M_1^{\varphi_1}}\) is diffuse. We first show that any intermediate subalgebra \({M_1 \subset Q \subset M}\) which has nontrivial central sequences in M is necessarily equal to M 1. Then we obtain a general structural result for all the intermediate subalgebras \({M_1 \subset Q \subset M}\) with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III1 factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.

Normalizers inside Amalgamated Free Product von Neumann Algebras

Recently, Adrian Ioana proved that all crossed products L∞(X)o (Γ1 ∗Γ2) by free ergodic probability measure preserving actions of a nontrivial free product group Γ1 ∗ Γ2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer NM (A)′′ of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M1 ∗B M2. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in L∞(X) o (Γ1 ∗ Γ2).

Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras

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.

Amalgamated free product type III factors with at most one Cartan subalgebra

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{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 $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

Free product von Neumann algebras associated to graphs, and Guionnet, Jones, Shlyakhtenko subfactors in infinite depth

Given a subfactor planar algebra P, Guionnet, Jones and Shlyakhtenko give a diagrammatic construction of a II1 subfactor whose planar algebra is P. They showed that if P is finite-depth, then the factors are interpolated free group factors, and they identified the parameters. We prove that if P is infinite-depth, then the factors are isomorphic to L(F∞).

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.

Factoriality, type classification and fullness for free product von Neumann algebras

We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras.

Examples of Prime von Neumann Algebras

Let {(Mi,φi) : i = 1, 2, . . .} be a family of injective von Neumann algebras on separable Hilbert spaces with a faithful normal state φi on each Mi and M be the reduced free product von Neumann algebra of (Mi,φi), i ∈ N. If there is a normal conditional expectation from M onto a non-injective von Neumann subalgebra N of M, then N is prime, i.e., N = N1⊗N2 implies that either N1 or N2 is a von Neumann algebra of type I. This result provides many examples of prime von Neumann algebras. These examples of prime von Neumann algebras include prime factors given by Ge (type II1) and by Shlyakhtenko (Type III). In our proof we combine Ozawa’s new techniques for solid von Neumann algebras with Shlyakhtenko’s “matrix model” techniques for the free ArakiWoods factors.

On spectra and Brown's spectral measures of elements in free products of matrix algebras

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})$, respectively. We prove that $AB$ is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0$. We show that if $X=AB$ or $X=A+B$ and $A,B$ are not scalar matrices, then the Brown measure of $X$ is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if $X=AB$ or $X=A+B$ and $X\neq \lambda 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$.

Every compact group arises as the outer automorphism group of a II 1 factor

We show that any compact group can be realized as the outer automorphism group of a factor of type II 1. This has been proved in the abelian case by Ioana, Peterson and Popa [A. Ioana, J. Peterson, S. Popa, Amalgamated free products of w-rigid factors and calculation of their symmetry group, math.OA/0505589, Acta Math., in press] applying Popa's deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs.