On the Classification of Full Factors of Type III

A characterization of fullness of continuous cores of type III$_1$ free product factors

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$_1$.

Classification of hyperfinite factors up to completely bounded isomorphism of their preduals

In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is semifinite and $\N$ is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III$_0$-factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors $\M$ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III$_1$-factor. In contr…

III型张量积因子的满性与Connes的$\boldsymbolτ$不变量

Fullness and Connes' $\boldsymbolτ$ invariant of type III tensor product factors

本文证明了任意两个满因子的张量积仍为满因子，计算了其Connes $τ$不变量，揭示了III型满因子的增强与最优谱隙性质，并证明了其与非I型顺从因子张量积的唯一McDuff分解。

On classification of non-unital amenable simple C*-algebras, III, the range and the reduction

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group which unifies the unital and nonunital, as well as stably projectionless cases. We also show that, for any given such invariant set, there is a finite separable simple $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algeb…

A Classification of Flows on AFD Factors with Faithful Connes--Takesaki Modules

We classify flows on AFD factors with faithful Connes-Takesaki modules. This is a generalization of classification of trace-scaling flows on the AFD $\mathrm{II}_\infty$ factor, which is equivalent to the uniqueness of the AFD $\mathrm{III}_1$ factor. In order to do this, we show that a flow on an AFD factor with faithful Connes-Takesaki module has the Rohlin property, which gives a partial answer to a characterization problem of the Rohlin property posed by Masuda-Tomatsu. It is also possible to think of this result as an $\mathbf{R}$-version of Izumi's result about compact group actions on t…

内射III型因子的正则子代数分类

Classification of regular subalgebras of injective type III factors

本文研究内射III型因子中满足相对交换子条件的正则子代数，利用相关顺从离散测度群胚及其在权流上的作用完成完整分类，并构造相应模型作用。

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

We investigate factoriality, Connes' type ${\rm III}$ 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 ${\rm III}$ invariants.

Rohlin flows on von Neumann algebras

We will introduce the Rohlin property for flows on von Neumann algebras and classify them up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II$_1$ factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III$_0$ factors. Several concrete examples are also studied.

III₁型因子的新构造

A new construction of factors of type ${\rm III_1}$

本文构造了一族新的III₁型因子，并在一定条件下借助Popa的结果给出完全分类，同时构造出具有相同τ不变量但两两不同构的不可数III₁型因子族。

非奇异伯努利交叉积的分类结果

Classification results for nonsingular Bernoulli crossed products

本文针对自由群及其自由积群的非奇异伯努利作用，证明了 III 型因子的刚性与分类结果，首次在保概率测度情形之外给出无法用 Connes τ-不变量区分的大量互不同构 III$_1$ 型伯努利交叉积。

Spectral gap characterization of full type III factors

We give a spectral gap characterization of fullness for type $\mathrm{III}$ factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if $M$ is a full factor and $σ: G \rightarrow \mathrm{Aut}(M)$ is an outer action of a discrete group $G$ whose image in $\mathrm{Out}(M)$ is discrete then the crossed product von Neumann algebra $M \rtimes_σG$ is also a full factor. We apply this result to prove the following conjecture of Tomatsu-Ueda: the continuous core of a type $\mathrm{III}_1$ factor $M$ is full if and on…

On the classification problem for C*-algebras

In the given article it is introduced new notions of a C$^*$-algebra of von Neumann type I and C$^*$-algebras of types I$_n$, II, II$_1$, II$_\infty$ and III. It is proved that any GCR-algebra is a C$^*$-algebra of von Neumann type I, and a C$^*$-algebra is an NGCR-algebra if and only if this C$^*$-algebra does not have a nonzero Abelian annihilator. Also an analog of the theorem on decomposition of a von Neumann algebra to subalgebras of types I, II and III is proved. In the final part it is proved that every C$^*$-factor of von Neumann type I is a C$^*$-algebra of type I$_n$ for some cardina…

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.

Full factors, bicentralizer flow and approximately inner automorphisms

We show that a factor $M$ is full if and only if the $C^*$-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type $\mathrm{III}_1$ factor is always ergodic. As a consequence, for any type $\mathrm{III}_1$ factor $M$ and any $λ\in ]0,1]$, there exists an irreducible AFD type $\mathrm{III}_λ$ subfactor with expectation in $M$. Moreover, any type $\mathrm{III}_1$ factor $M$ which satisfies $M \cong M \otimes R_λ$ for some $λ\in ]0,1[$ has trivial bicentralizer. Finally, we give a counter-example to the charact…

Partial classification of the Baumslag-Solitar group von Neumann algebras

We prove that the rational number |n/m| is an invariant of the group von Neumann algebra of the Baumslag-Solitar group BS(n,m). More precisely, if L(BS(n,m)) is isomorphic with L(\BS(n',m')), then |n'/m'| = |n/m| or |m/n|. We obtain this result by associating to abelian, but not maximal abelian, subalgebras of a II_1 factor, an equivalence relation that can be of type III. In particular, we associate to L(BS(n,m)) a canonical equivalence relation of type III_|n/m|.

具有最小外自同构群的 III$_1$ 型因子

A type III$_1$ factor with the smallest outer automorphism group

本文构造了一个具有可分预对偶的满的 III$_1$ 型因子，其外自同构群最小，即典范模同构嵌入是同构，方法为融合自由积。

On the classification and modular extendability of E$_0$-semigroups on factors

In this paper we study modular extendability and equimodularity of endomorphisms and E$_0$-semigroups on factors with respect to f.n.s. weights. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. We say that a modularly extendable E$_0$-semigroup is of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We also compute the coupling index and the relative commutant index for the CAR flow…

II$_1$ 因子上的流与 Connes 双中心化子问题的分类

The classification of flows on $\mathrm{II}_1$ factors and Connes' bicentralizer problem

本文解决了超有限 II$_1$ 因子上流的分类问题，并利用新发现的共振现象证明了所有 III$_1$ 型因子的 Connes 双中心化子猜想。

The classification problem for von Neumann factors

We prove that it is not possible to classify separable von Neumann factors of types $\II_1$, $\II_\infty$ or $\III_λ$, $0\leq λ\leq1$, up to isomorphism by a Borel measurable assignment of "countable structures" as invariants. In particular the isomorphism relation of type $\II_1$ factors is not smooth. We also prove that the isomorphism relation for von Neumann $\II_1$ factors is analytic, but is not Borel.

Non-classification of free Araki-Woods factors and $τ$-invariants

We define the standard Borel space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $τ$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.

纠缠挪用、量子场与冯·诺依曼代数的分类

Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras

本文在冯·诺依曼代数框架下研究纠缠挪用任务，定义了两个代数不变量，证明类型 III$_1$ 因子是普适挪用者，并据此解释量子场论中 Bell 不等式的最大违背。

Solidity of type III Bernoulli crossed products

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…

绝缘体的拓扑分类：III. 所有维度下的非相互作用谱间隙系统

Topological Classification of Insulators: III. Non-interacting Spectrally-Gapped Systems in All Dimensions

本文通过定义哈密顿量空间及其拓扑，利用强拓扑不变量作为完全不变量，推导出Kitaev周期表，并确认了谱间隙区域拓扑相与阿贝尔群之间的一一对应关系。