A Kurosh-Type Theorem for Type III Factors

真无限因子中的Voiculescu定理

Voiculescu's Theorem in Properly Infinite Factors

本文研究可分真无限因子中的Voiculescu近似酉等价定理，给出可约算子范数稠密性、广义双交换子定理等应用，将结果推广到可分$\text{III}$型因子乘子代数，并引入核长度概念。

Unitary conjugacy for type III subfactors and W$^*$-superrigidity

Let $A,B\subset M$ be inclusions of $σ$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition $A\preceq_MB$ using their modular actions. In the main theorem, we prove that if $A\preceq_MB$ holds, then an intertwining element for $A\preceq_MB$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A\preceq_MB$ in terms of their continuous cores. Using this new characterization, we prove the first W$^*$-superrigidity type result for grou…

Factors of type III and the distribution of prime numbers

This version of the paper corrects an inaccuracy in the proof of Theorem 2.9 in the published version. The main results remain unchanged.

Fixed point and spectral characterization of finite dimensional C*-algebras

We show that the following conditions on a C*-algebra are equivalent: (i) it has the fixed point property for nonexpansive mappings, (ii) the spectrum of every self adjoint element is finite, (iii) it is finite dimensional. We prove that (i) implies (ii) using constructions given by Goebel, that (ii) implies (iii) using projection operator properties derived from the spectral and Gelfand-Naimark-Segal theorems, and observe that (iii) implies (i) by Brouwer's fixed point theorem.

Classification of Roberts actions of strongly amenable C^* tensor categories on the injective factor of type III_1

In this paper, we generalize Izumi's result on uniqueness of realization of finite C$^*$-tensor categories in the endomorphism category of the injective factor of type II_1 for finitely generated strongly amenable C$^*$-tensor categories by applying Popa's classification theorem of strongly amenable subfactors of type III_1.

Classification of actions of discrete amenable groups on strongly amenable subfactors of type III$λ$

Using the continuous decomposition, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III$_λ, 0 < λ< 1$. Winsløw's fundamental homomorphism is a complete invariant. This removes the extra assumptions in the classification theorems of Loi and Winsløw and gives a complete classification up to cocycle conjugacy.

The Fine Structure of the Kasparov Groups III: Relative Quasidiagonality

In this paper we identify QD(A,B), the quasidiagonal classes in KK_1(A,B), in terms of K_*(A) and K_*(B), and we use these results in various applications. Here is our central result. Theorem: Suppose that A is in the category of separable nuclear C^*-algebras which satisfy the UCT and A is quasidiagonal relative to B. Then there is a natural isomorphism QD(A,B) = Pext (K_*(A), K_*(B))_0 . Thus quasidiagonality of KK-classes is indeed a topological invariant. We give several applications. Finally, we establish a converse to a theorem of Davidson, Herrero, and Salinas, giving conditions under w…

Fourier analysis for type III representations of the noncommutative torus

For the noncommutative 2-torus, we define and study Fourier transforms arising from representations of states with central supports in the bidual, exhibiting a possibly nontrivial modular structure (i.e. type III representations). We then prove the associated noncommutative analogous of Riemann-Lebesgue Lemma and Hausdorff-Young Theorem. In addition, the $L^p$- convergence result of the Cesaro means (i.e. the Fejer theorem), and the Abel means reproducing the Poisson kernel are also established, providing inversion formulae for the Fourier transforms in $L^p$ spaces, $p\in[1,2]$. Finally, in $…

Category of Noncommutative CW complexes. III

We prove in this paper a noncommutative version of Leray Spectral Sequence Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with $\E^2$ terms as $\E^2_{p,q} = \HP_p(A; \HP_q(B,A)) \Longrightarrow \HP_{p+q}(B)$ and $\E^2_{p,q} = \HP_p(A;\K_q(B,A)) \Longrightarrow \K_{p+q}(B)$.

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…

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…

强遍历等价关系：谱隙与III型不变量

Strongly ergodic equivalence relations: spectral gap and type III invariants

本文利用谱隙判据刻画了标准测度空间上的强遍历等价关系，并证明了局部紧阿贝尔群可测1-上闭链导出的斜积等价关系的强遍历性，同时引入并研究了III型强遍历等价关系的Sd和τ不变量及其结构定理。

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…