Dimitri Shlyakhtenko
We show that for each (0<λ<1), the free Araki-Woods factor of type III(_λ) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime), and does not contain a Cartan subalgebra.
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We prove some unique prime factorization results for tensor products of type $II_1$ factors of the form $Γ_q(\mathbb{C}, S \otimes H)$ arising from symmetric independent copies with sub-exponential dimensions of the spaces $D_k(S)$ and dim$(H)$ finite and greater than a constant depending on $q$.
This version of the paper corrects an inaccuracy in the proof of Theorem 2.9 in the published version. The main results remain unchanged.
In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse free product components. In the type III setting, this is the first classification result for infinite tensor product non-amenable factors. Our proof is based on Popa's intertwining techniques and a characterization of relative amenability on the continuous cores.
Unique prime factorization and bicentralizer problem for a class of type III factors
针对包含自由 Araki-Woods 因子的大类von Neumann代数，证明了张量积的唯一素分解及Connes双中心化子问题的正解。
We prove some unique factorization results for tensor products of free quantum group factors. They are type III analogues of factorization results for direct products of bi-exact groups established by Ozawa and Popa. In the proof, we first take continuous cores of the tensor products, which satisfy a condition similar to condition (AO), and discuss some factorization properties for the continuous cores. We then deduce factorization properties for the original type III factors. We also prove some unique factorization results for crossed product von Neumann algebras by direct products of bi-exac…
We investigate the ideal structures of the C^*-algebras arising from topological graphs. We give the complete description of ideals of such C^*-algebras which are invariant under the so-called gauge action, and give the condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our C^*-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph such that the associated C^*-algebra is prime but not primitive.
In this paper we provide new examples of property (T) group factors with trivial fundamental group thereby providing more evidence towards Popa's conjecture on triviality of fundamental groups for property (T) group factors (page 9 \cite{Po13}; see also Problem 2, page 551 in Connes' book \cite{Co94}). Our groups arise as direct product of groups either in Class $\mathscr S$ or Class $\mathscr V$, as introduced in \cite{CDHK20}. We establish a unique prime factorization result for these product groups, thereby providing more evidence towards Popa's conjecture on prime decomposition of property…
We prove several unique prime factorization results for tensor products of type II_1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if $R \otimes LF_{r_1} \otimes ... \otimes LF_{r_m}$ is isomorphic to a subfactor in $R \otimes LF_{s_1} \otimes >... \otimes LF_{s_n}$, for some $2\leq r_i, s_j \leq \infty$, then $m\le n$.
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 give a new proof of a theorem due to Alain Connes, that an injective factor $N$ of type III$_1$ with separable predual and with trivial bicentralizer is isomorphic to the Araki--Woods type III$_1$ factor $R_{\infty}$. This, combined with the author's solution to the bicentralizer problem for injective III$_1$ factors provides a new proof of the theorem that up to $*$-isomorphism, there exists a unique injective factor of type III$_1$ on a separable Hilbert space.
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.
We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $φ_0$ and any discrete group $Γ$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,φ_0)^{\mathbin{\bar{\otimes}}Γ}\rtimes Γ$ is solid relatively to $\mathcal{L}(Γ)$. In particular, if $\mathcal{L}(Γ)$ is solid then $M$ is solid and if $Γ$ is non-amenable and $A_0 \neq \mathbb{C}$ then $M$ is a full prime factor. This gives many new examples of solid or prime type $\mathrm{III}$ factors. Following Chifan and Ioana, we also obta…
We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type III$_1$.
Type III_1 factors arising as (direct summands of) von Neumann algebraic free products are investigated. In particular we compute Connes' Sd- and tau- invariants for those type III_1 factors without any extra assumption.
We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. Our main technical tools are the structural analysis of type III factors and the theory of canonical extension of endomorphisms introduced by Izumi.
We use malleable deformations combined with spectral gap rigidity theory, in the framework of Popa's deformation/rigidity theory to prove unique tensor product decomposition results for II$_1$ factors arising as tensor product of wreath product factors. We also obtain a similar result regarding measure equivalence decomposition of direct products of such groups.
We introduce a new invariant \mathcal{S}(M) for type III factors M with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant τcannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
We prove that, for any type III$_1$ free product factor, its continuous core is full if and only if its $τ$-invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free Araki--Woods factors. Moreover, our method shows the same result for full (generalized) Bernoulli crossed product factors of type III$_1$.
We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type ${\rm II}_{1}$ factor representations are used in these realizations.
In this last article of the series on outer actions of a countable dicrete amenable group on AFD factors, we analyze outer actions of a countable discrete free abelian group on an AFD factor of type $\text{III}_λ$, $0<λ< 1$, and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a $\Bbb T$-valued cocycle of a torsion free abelian group, which might have been known by the group cohomologists. As the asymmetrization technique gives us a very handy criteria for coboundaries, we present it here in detail in the second section.
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 prove that any II$_1$ factor arising from a free ergodic probability measure preserving action $Γ\curvearrowright X$ of a product $Γ=Γ_1\times\dots\timesΓ_n$ of icc hyperbolic, free product or wreath product groups is prime, provided $Γ_i\curvearrowright X$ is ergodic, for any $1\leq i\leq n.$ We also completely classify all the tensor product decompositions of a II$_1$ factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II$_1$ factors. Finally…
We prove that if $Γ$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(Γ)$ is prime. In particular, we deduce that the II$_1$ factors associated to the arithmetic groups $\text{PSL}_2(\mathbb Z[\sqrt{d}])$ and $\text{PSL}_2(\mathbb Z[S^{-1}])$ are prime, for any square-free integer $d\geq 2$ with $d\not\equiv 1 mod{4}$ and any finite non-empty set of primes $S$. This provides the first examples of prime II$_1$ factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe…
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