M. Caspers, D. Potapov, F. Sukochev
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).
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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…
We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\hbox{II}_1$, $\hbox{II}_\infty$, $\hbox{III}_λ$, $0 < λ\leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space).
Uniqueness of almost periodic outer flows on the hyperfinite type $\mathrm{II}_1$ factor
本文研究超有限型II₁因子上几乎周期外流的唯一性，证明其满足Rokhlin性质并唯一到余cocycle共轭。
Using methods of R.J.Tauer we exhibit an uncountable family of singular masas in the hyperfinite $\textrm{II}_1$ factor $\R$ all with Pukánszky invariant $\{1\}$, no pair of which are conjugate by an automorphism of $R$. This is done by introducing an invariant $Γ(A)$ for a masa $A$ in a \IIi factor $N$ as the maximal size of a projection $e\in A$ for which $A e$ contains non-trivial centralising sequences for $eN e$. The masas produced give rise to a continuous map from the interval $[0,1]$ into the singular masas in $\R$ equiped with the $d_{\infty,2}$-metric. A result is also given showing …
Almost almost periodic type $\mathrm{III}_1$ factors and their 3-cohomology obstructions
本文构造了外模流几乎周期但无几乎周期态的全因子，揭示了非平凡整二次关系导致的3-上同调障碍，并证明了超有限$\mathrm{II}_1$因子及交叉积因子的相关正面结果。
We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group $B_0^{\mathbb N}$. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type ${\rm I}_\infty$ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,1…
We study the von Neumann algebra, generated by the regular representations of the infinite-dimensional nilpotent group $B_0^{\mathbb Z}$. In [14] a condition have been found on the measure for the right von Neumann algebra to be the commutant of the left one. In the present article, we prove that, in this case, the von Neumann algebra generated by the regular representations of group $B_0^{\mathbb Z}$ is the type ${\rm III}_1$ hyperfinite factor. We use a technique, developed in [20] where a similar result was proved for the group $B_0^{\mathbb N}$. The crossed product allows us to remove some…
We investigate $E_0-$semigroups on general factors, which are not necessarily of type I, and analyse associated invariants like product systems, super product systems etc. By tensoring $E_0-$semigroups on type I factors with $E_0-$semigroups on type II$_1$ factor, we produce several families (both countable and uncountable), consisting of mutually non-cocycle-conjugate of $E_0-$semigroups on the hyperfinite II$_\infty$ factor. Using CCR representations associated with quasi-free states, we construct for the first time, uncountable families consisting of mutually non-cocycle-conjugate $E_0-$sem…
We examine the notion of $α$-strong singularity for subfactors of a \IIi factor, which is a metric quantity that relates the distance between a unitary in the factor and a subalgebra with the distance between that subalgebra and its unitary conjugate. Through planar algebra techniques, we demonstrate the existence of a finite index singular subfactor of the hyperfinite \IIi factor that cannot be strongly singular with $α=1$, in contrast to the case for masas. Using work of Popa, Sinclair, and Smith, we show that there exists an absolute constant $0<c<1$ such that all singular subfactors are $c…
Anomalies for conformal nets associated with lattices and $T$-kernels
本文在超有限因子上构造偶格对应环面的$T$-核，计算其Sutherland阻碍类，并将其与共形网扭曲扇区的反常及格的内积联系起来。
The undecidability of having the QWEP
该研究证明具有QWEP性质的C*-代数类与W*-概率空间类均无法有效公理化，还得出超有限III₁因子、带典范Powers态的Powers因子ℛ_λ（0<λ<1）不具备可计算通用理论的结论。
A Note on Pseudofinite W*-Probability Spaces
引入伪有限W*-概率空间，证明它们是因子且非III_0型，构造显式III_λ因子例子，证明伪有限因子是满的，推广了Farah-Hart-Sherman定理。
Structure of bicentralizer algebras and inclusions of type III factors
本文研究了类型III1子因子包含的相对双中心化子代数结构，构造了与态选择无关的典范流，并证明了相对双中心化子定理，在顺从情形下解决了Kadison问题。
The paper shows deep connections between exotic smoothings of small R^4, noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative C*-algebras we show that cyclic cohomology invariants characterize some small exotic R^4. Certain exotic smooth R^4's define a generalized embedding into a space which is K-theoretic equivalent to a noncommutative Banach algebra. Furthermore, we show that a factor III von Neumann algebra is naturally related with nonstandard smoothing of a small R^4 and conjecture that this factor is the unique hy…
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