A Note on Approximate Liftings

$C^*$-代数的提升性质：从局部到全局？

The lifting property for $C^*$-algebras : from local to global ?

针对 Kirchberg 关于 $C^*$-代数中 LLP 蕴含 LP 的猜想，本文给出若干针对 WEP/QWEP 情形的等价判据，并联系可控有限维算子空间、$C^*$-张量积与局部自反性变体，但未能判定其是否成立。

局部提升性质、FD性质和近似表示的稳定性

The Local Lifting Property, Property FD, and stability of approximate representations

本文研究了3维流形群、极限群及某些一关系群和直角Artin群的局部提升性质和FD性质，并证明这些群在归一化单位不变范数下具有高度灵活性的稳定性。

渐近同伦提升性质

An asymptotic homotopy lifting property

本文提出了渐近同伦提升性质，并证明该性质对半射影$C^*$-代数的序列归纳极限等可分$C^*$-代数成立，且在商映射满足近似分解性质时对任意可分$C^*$-代数亦成立。

A Note on Approximately Divisible C$^*$-algebras

Let $\mathcal A$ be a separable, unital, approximately divisible C$^*$-algebra. We show that $\mathcal A$ is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of $\mathcal A$ is less than or equal to 1. In addition, we show that the similarity degree of $\mathcal A$ is at most 5. Thus an approximately divisible C$^*$-algebra has an affirmative answer to Kadison's similarity problem.

A note on the Akemann-Doner and Farah-Wofsey constructions

We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable $C^*$-algebra $A$ with only separable commutative $C^*$-subalgebras. We also extend a result of Farah and Wofsey's, constructing $\aleph_1$ commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin's almost disjoint family construction.

关于强相似性及Connes嵌入问题的注记

A note on strong similarity and the Connes embedding problem

本文研究了强相似性与Connes嵌入问题，证明了某些完全有界同态无法被正交化，且不具有特定提升性质。

从稳定秩一到实秩零：关于迹近似振荡零的注记

From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero

本文通过迹振荡方法证明了稳定秩一的C*-代数具有实秩零，并建立了迹近似振荡零与实秩零之间的等价关系。

无提升性质的四维算子系统

Four-dimensional operator systems without the lifting property

构造了Calkin代数中无单位完全正提升的四维算子系统，推广到任意n≥2，并给出广义Smith-Ward问题的反例。

II1因子中N-独立集的提升

Lifting for N-independent sets in II1 factors

本文研究II1因子中的N-独立集，证明近似2N-独立的有限元组可通过算子范数扰动变为精确N-独立，从而推广相关定理并解决开放问题。

A note on the Voiculescu's theorem for commutative C$^*$-algebras in semifinite von Neumann algebras

In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.

关于FLC点集的内顺从性注记

A note on inner amenability for FLC point sets

本文证明第二可数局部紧群中近似格关联的群胚是内顺从的，并将结果推广到有限局部复杂性点集，部分回应了Ananantharman-Delaroche关于étale群胚内顺从性的公开问题。

Full quantum crossed products, invariant measures, and type-I lifting

We show that for a closed embedding $\mathbb{H}\le \mathbb{G}$ of locally compact quantum groups (LCQGs) with $\mathbb{G}/\mathbb{H}$ admitting an invariant probability measure, a unitary $\mathbb{G}$-representation is type-I if its restriction to $\mathbb{H}$ is. On a related note, we also prove that if an action $\mathbb{G}\circlearrowright A$ of an LCQG on a unital $C^*$-algebra admits an invariant state then the full group algebra of $\mathbb{G}$ embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). W…

动力扰动与$C^*$-代数提升问题

Dynamical Perturbing and $C^*$-algebra Lifting Problems

本文研究了近似群作用的稳定性问题，并引入了新的$C^*$-代数提升问题，探讨了有限群作用的稳定性及条件半射影性。

A note on geodesics of projections in the Calkin algebra

Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $π:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $π(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal…

Order theory and interpolation in operator algebras

We continue our study of operator algebras with and contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algeb…